(z) E a. From (6.64) we have (6.65) Consider a point Wo of the domain Po< 1wi < +00. In (6.65) setting w =Wo and letting Il-+oo, then gm, (wo)-G(wo), and from the inequality 1C1l-.{gm.(Wo)} -c1>{G(wo)}I::::;; 1C1l-.{g .. .cwo)} -c1>{g ... (wo)}1 + 1c1>{g .. .c w o)} - c1>{G(wo)} 1
Normal F amihes of Meromorp hie Functwns
264 we see that
Consequently G(wo) = Wo -
!l){G(wo)}
and in general we have G(w) = w -
!l){G(w)}
in the domain Po
E ~ and the proof of Theorem 6. 15 is com-
plete. Consider a function g (w) E~. By (6. 63), g (w )-=, as w-=, and then by (6.62), in the domain Po< Iwi < +=, we have 9 (w) = w
+ ljJ(w),
(6.66)
where 1/1(w) is a holomorphic function in the domain Po< Iwi < += with lim 1/1 (w) = O. Consequently the Laurent series expansion of 1/1 (w) in the domain Po< I w 1<+= has the following form:
=
ljJ(w) =
2:
(6.67)
c'••
a=l W
As for the coefficient b n in (6.53), we see that for each
n~I,
the coefficient
Cn
in (6.67) is a continuous functional defined on~. Of course this also holds for Icnl. Note that we can also consider en as a functional defined on u.
Theorem 6. 16. Let D be a domain and ZI,Z2(z66 z2) two points of D. Let $T
be the family of functions f(z) satisfying the following conditions: 1 ° f (z) is merom orphic and univalent in D. 2°f(zl)=O,
fl
(zl)=I; f(Z2)=='
265
Further Applications Then the family $T is closed in D.
Proof. First let us show that if fCz) E $T, then Z2 is a simple pole of fCz). In fact, consider a circle c: Iz-z21
=
(z -
z2)f •• (z) (k
= 1,2,···)
holomorphic in D, which converges locally uniformly in D. to (z-z2)F(z). Let
f: Iz-z21 ~p be a circle which belongs to D and does not contain the point z!. Since on the circle Iz-z21 =p, g. (z) converges uniformly to (z-z2)F (z), hence g. (z) converges uniformly to a holomorphic function g (z) in the circle
r:
IZ-Z2
I
z2)F(z)
for 0< Iz-z21 <po Furthermore since g. (z) has no zero in
r.
r , g (z)
has no zero in
So for 0< IZ-Z21
F (z)
=
..J..i!l z - Z2
and Z2 is a simple pole of F(z). Consequently F(z) E$T, provided that we define F(Z2)==' Since fn,(z)-F(z) uniformly on the circle IZ-Z21 =p and the functions fn, (z) and F (z) have no zero in the circle
f, we see that 1/fn, (z)-l!F (z) uniform-
Normal Families of Meromorphic Functions
266 ly in
f. Thus the sequence fn,. (z) (k = 1,2,,,,) converges locally uniformly in D. to
F (z) and the sequence 11fn,. (z) (k = 1, 2, ••• ) converges uniformly to 11F (z) in
f. This implies that the sequence fn,. (z) (k = 1,2,,,,) converges locally uniformly in D to F (z), with respect to the spherical distance. Hence the family
.sr is
closed in D according to Definition 4. 5.
Corollary 6. 12. Given a number O
°
1 f{z) is meromorphic and univalent in the circle C: Iz I <1. 2 0 f(0)=0, fl (0)=1; f(q)=oo. Then the family S(q) is closed in the circle C. Consider the domain DJ=C-(q) in which the functions of the family S(q) are holomorphic.
A fortiori,
S (q) is closed in DJ according to Definition 4. 5, and
hence according to Definition 4. 1, by Lemma 4.3. It follows that,
m~O
being
an integer and Zo a point of D J , f(m) (zo) and If(m) (zo) I are continuous functionals defined on S (q). On the other hand, by the remark following Definition 4. 5, the family Y(q) = (g(z) = 1/i(z) If{z) E S(q)} is closed in C. Similarly we see that m~O being an integer and Zo a point of the domain C - (0), (l/f(z) }~~;, and I (l/f{z) }~~;, I are continuous functionals defined on S(q). Since each function f(z) ES(q) has a simple zero at z= 0, we can apply Theorem 4. 11 to the family S (q) with .1, = ( I w I < t) (0< t <
+ 00) and conclude
that there exists a circle [': I w I lit) (O
Q:
Iw I >R (O
00) such that for each function f(z) ES (q), the values of f(z) cover Q.
N ow we treat the case of multivalent functions. First give the following definition:
Definition 6. 2.
p~2 being an integer, a meromorphic function f(z) in a
Further Appilcations
267
domain D is said to be p-valent in D. if the following conditions are satisfied: 1° For each value wE C the equation f(z) =w has at most p roots in D. with due count of order of multiplicity. 2° There exists at least one value wEC such that the equation f(z)=w has p roots in D.
Theorem 6. 17. Let D be a domain. Zo a point of D and p?2 an integer. Let $T be the family of the functions Hz) satisfying the following conditions: 1° f(z) is holomorphic and p-valent in D. 2° f(j)(zo) =0 (j=O.l.···.p-1). f(p)(zo)=1. Then the family $T is closed in D.
Proof. As in the proof of Theorem 6. 12. we first show that given any sequence of functions fn (z) (n = 1. 2 •••• ) of the family $T. we can extract a subsequence fn,. (z) (k = 1. 2 •••• ) which converges locally uniformly to a holomorphic function f(z) in D. with f(j)(zo) = 0 (j = O.I.···.p -
1) .f(')(zo) = 1.
It remains to show that the function f(z) is p-valent in D. In fact consider a fi-
nite value wand assume that the function f(z) -w has a zero Zl of order m in D. Let
r : IZ-Zl I =r be a circle such that f: IZ-Zl I~r belongs to D and that f(z)-
w has no zero for 0< I z - Zl I ~r. Since the sequence fn, (z) (k = 1.2 •••• ) converges uniformly to f(z) on
r • when k is sufficiently large. the function fn, (z)r • by a known theorem. It is then clear that the
w has m zeros in the interior of
function f(z) -w has at most p zeros in D. On the other hand. f(z) has a zero of order p at zoo Consequently the function f(z) is p-valent in D and hence belongs to the family $T. $T is therefore closed in D.
Corollary 6. 13. Given an integer p?2. let M (p) be the family of the functions f(z) satisfying the following conditions: 1° f(z) is hoI om orphic and p-valent in the circle Iz I <12°f(j)(0)=0 (j=O.l.···.p-1). f(p)(O)=1. Then the family M(p) is closed in the cricle Izl<1. Applying Theorem 4. 11 to the family M (p). we see that there is a circle [':
Normal Fam,l,es of Meromorphic Functions
268
Iwi
r,
where p depends on p.
Theorem 6. 18. Let D be a domain, Zl ,Z2(ZI#Z2) two points of D and p~ 2 an integer. Let $I' be the family of the functions f(z) satisfying the following conditions: 1° f(z) is merom orphic and p-valent in D.
2°
fW
(Zl) = 0 (j=O, 1, ••• ,p-l), f(p)(zl)= l.
3° Z2 is a pole of order p of f(z). Then the family $I' is closed in D. With the help of Theorem 6. 17, the proof of Theorem 6.18 is quite similar to that of Theorem 6.16. Corollary 6. 14. Given an integer p~2 and a number 0<1;<1, let M (p, I;) be the family of the functions f(z) satisfying the following conditions:
1 ° f(z) is merom orphic and p-valent in the circle Iz I (0)=0 (j=O,I,···,p-l), f(p)(O)=l. 3° I; is a pole of order p of f(z). Then the family M (p ,I;) is closed in the circle Iz I R. Up to now we have shown that some families of univalent functions or mUltivalent functions are closed families. However besides these families, we can define more general families of such functions, which are also closed. For this we give only some examples in the case of holomorphic functions.
Theorem 6. 19. Let D be a domain, E a bounded closed set of points belonging to D and a>O,I3>O(a
269
Further App hcations
Proof. Let fn(z) (n=1,2, .. ·) be a sequence of functions of the family Y. From the sequence of points 1;,.,=1;,(fn) (n= 1,2,,,,) we can extract a subsequence 1;,,,, = 1;, (f", ) (k = 1,2,,,,) which converges to a point ZE E. Consider a circle c: Iz -zl~o belonging to D. Let ko be a positive integer such that for k~ko,
/;.n, be-
longs to the circle c: Iz-z I <0. Then the family of functions {fo,. (z) (k~ko)} is normal in the domain D1 =D-c. Adding the finite number of functions fo,. (z) (1 ~k~ko-l)
to the family {fo,. (z) (k~ko)}, we get the family of functions {fo,.
(z) (k~l)} which is also normal in DI' Since 0 can be taken arbitrarily small so the family {fn, (z) (k~l)} is normal in the domain Do=D- (Z). Consequently we can extract from the sequence fo,. (z) (k = 1,2,,,,) a subsequence fm, (z) (1.= 1 ,2, ... ;mA = nt) which converges locally uniformly in Do to a holomorphic function F(z) or to
00.
We are going to show that the second case can not occur. In
fact consider the sequence of functions
g,,(z)
=
fm, (z) (z _
Sm) (), =
1,2,,,,)
which are holomorphic and have no zero in D. Let
f: Iz - Z I~r be a circle be-
longing to D. Let 1.0 be an integer such that for I.~I.o, l1;,m, -zi
on the circle I z - Z I = r. Consequently if the second case occurs, then the sequence gA (z) (I. = 1, 2, ... ) converges uniformly to
00
on the circle I z - Z I = r ,
and hence also in the circle T. But
so the second case can not occur. It is then clear that the sequence fm, (z) (1.= 1, 2 .... ) converges locally uniformly to a hoi om orphic function F (z) in D, and it is easy to see that F(Z)
= 0,
a~ IF' (Z)
I ~ p.
270
Normal Famtltes of Meromorphtc Functions
Thus F (z) E $T and the proof of Theorem 6. 19 is complete.
Theorem 6. 20. Let D be a domain, m~p)
p~2 an integer, Zj(j= 1,2, "',m; 1~
m distinct points of D and a>O, 13>0 (a
family of the functions satisfying the following conditions: 1 ° f(z) is holomorphic and p-valent in D. 2° For each l~j~m, z; is a zero of order A;=A;(f) of f(z), with a~ If(A,)(z)
1~13 and ~ A;=P' j~1
Then the family $T is closed in D.
Proof. Let fn (z) (n = 1,2, ••• ) be a sequence of functions of the family $T. It is easy to see that there is a system of posiitve integers Aj(j= 1,2, ••• ,m) with L Aj=P such that for an infinite number of positive integers n, we have j~1
So we can find a subsequence fn, (z) (k = 1,2, ••• ) of the sequence fn (z) (n = 1,2, ••• ) such that
A/f.)
= Aj
(j = 1,2,···,m).
The family of functions (fn,(Z) (k=1,2,···)} is normal in the domain D1=D(Z1 ,zz, •.• ,zm) and hence we can extract from the sequence fn, (z) (k = 1,2, ••• ) a subsequence fq,(z) (s=1,2,···;q,=n.,> which converges locally uniformly in D1 to a holomorphic function F (z) or to
00.
By means of the sequence of functions
and the condition 2° in Theorem 6.20, we see, as in the proof of Theorem 6. 19, that the second case can not occur. Hence the sequence fq, (z) (s = 1,2, ••• ) converges locally uniformly in D to a holomorphic function F (z). It is easy to see
that F(z) satisfies the condition 2° in Theorem 6.20 and is p-valent in D. Thus
Further Apphcatwns
F(z) E :JT and Theorem 6.20 is proved.
271
7 EXTENSIONS OF SOME CRIITERIONS OF NORMALITY AND QUASI-NORMALITY
In this chapter we shall give some extensions of Theorems 2. 1.2. 8.3. 1.3. 9.5. 4 and Corollaries 5.9.5. 10. such that the values a. b.c and aj (j= 1. 2.3) in these theorems and corollaries are replaced by functions. The results thus obtained will then be applied to prove theorems on sets of functi.ons covered. in a certain sence. by the values of holomorphic or meromrophic functions. Similar theorems concerning univalent or multivalent functions will also be proved. by a different method.
7. 1. CASE OF HOLOMORPHIC FUNCTIIONS Theorem 7. 1. Let D be a domain and
M.o two positive numbers.
Let $
be the family of the functions f(z) satisfying the following conditions: l°lf(z) is holomorphic in D.
2 0 There are two holomorphic functi.ons
f
I/> (z ) I
(z) -
>
<5.
cp(z)
I cp(z) I
"# O. f
< M.
(z) -
II/> (z ) I
I/> (z)
"# 0
<M
(7.1)
(7.2)
in D. Then the family $
is normal in D.
Proof. Consider a sequence of functions fn (z) (n = 1. 2 •••• ) of the family $.
By the condition 2 0 • to each fn(z) correspond two functions \Po (z) and Wn (z)
273
274
Normal Famtltes of Meromorphtc Functions
holomorphic and satisfying the conditions (7. 1), (7. 2) in D, with respect to f n (z). The sequence of functions
g.
(z) = f.(z) 1jJ.(z) -
gJ.(z) (n = 1,2,"') gJ.(z)
are holomorphic in D and do not take the values 0 and 1. By Theorem 2. 1, the family {gn (z) (n = 1,2, ".)} is normal in D. Hence we can find a subsequence g .. (z) (k = 1,2, ••• ) converging locally uniformly in D either to a holomorphic function or to
00.
In the second case, we deduce from the relation f.,(z)
=
{1jJ.,(z) -
gJ.,(z)}g.,(z)
+ gJ.,(z)
and (7. 1) that fn,Cz) (k = 1, 2, ".) also converges locally uniformly to
00
in D.
In the first case, the same relation and (7. 1) show that the sequence fn, (z) (k = 1, 2, ".) is locally uniformly bounded in D, hence by Corollary 1. 3, we can extract from fn, (z) (k = 1,2, ".) a subsequence fm, (z) (s = 1,2, ".) converging locally uniformly to a hoi om orphic function in D.
Definition 7. 1. Let S, (i = 1,2) be two sets of holomorphic functions in a domain D. We say that S,(j=1,2) are locally uniformly disjoint, if to each point Zo E D correspond a circle Iz-zo I
1" M
such that for any two functions qlj(z)ESI(j=1,2), we have ICA (z) -
lP.!(z) I
> 1/,
Igj(z) 1< M (i = 1,2)
in the circle Iz-zol
Theorem 7.2. Let
sr be a family of holomorphic functions
in a domain D
and SI (i = 1,2) two locally uniformly disjoint sets of holomorphic functions in D. Suppose that to each point Zo E D corresponds a circle I z - Zo I
sr,
there are two func-
tions qlj(z)ESI(j=1,2) such that the two functions f(z)-qlj(z) (j=1,2) have no zero in the cricle Iz-zo I
sr is
normal in D.
275
ExtenSIons of Some Cnterions
Pr oof. Consider a point Zo E D. Let Iz -Zo I
IZ-Zo I
and two positive numbers 11, M having
the property in Definition 7. 1. Setting ro =min (r, r' ), then by Theorem 7. 1, the family
.JjfT
is normal in the circle
IZ-Zo I
Since Zo is arbitrary,
.JjfT
is nor-
mal in D by Theorem 1. 4.
Definition 7. 2. Let f(z) be a holomorphic function in a domain D and S a set of holomorphic functions in D. We say that the values of the function Hz) cover in generalized sense the set S, if for each function
Corollary 7. 1. Let D be a domain, E I(j=1,2) two bounded closed sets of points belonging to D and SI (j = 1,2) two locally uniformly disjoint sets of holomorphic functions in D. Then there exists a positive number b depending onlyon D,E l t S;(i=I,2) having the following property: If Hz) is a holomorphic function in D such that min If(z)
I:::;; 1,
max If(z)
I ~ b,
zEN:
zEE1
then the values of the function f(z) cover in generalized sense at least one of the sets S;0=1,2).
Proof. Assume that the positive number b does not exist. Then for each positive integer n, there is a holomorphic function fn (z) in D such that mtn If.(z) zEN]
I:::;; 1,
max If.(z)
I~n
(7. 3)
zEN:.
and that the values of the function fn (z) do not cover in generalized sense either of the sets S;(j=1,2). Accordingly there are two functions
Normal Families of Meromorphic Functions
276
Definition 7. 3. Let D be a domain in the z-plane and a a set of points of the w-plane. We denote by Ho(a) the set of holomorphic functions qJ(z) in D such that the values taken by qJ(z) in D all belong to a. In particular, each point wE a considered as a constant function defined in D belongs to Ho(a).
Corollary 7. 2. Let D be a domain and Ei G= 1,2) two bounded closed sets of points belonging to D. Then there exist three positive numbers 0.,0,1. depending only on D,E,(j=1,2) having the following property: If f(z) is a holomorphic function in D such that min If(z) 2:EB J
I ~ a,
max If(z) IZEE z
I ~ 1,
(7.4)
then the values of the function f(z) cover in generalized sense a set Ho (PI < Iw I 0, P2 -
PI
> A.
(7.5)
Proof. By Corollary 7.1, corresponding to D, E,(j=l ,2), SI=HoO< Iw 1<2), S2=H o (3< Iw 1<4), there is a positive number b having the property stated in that corollary. Then the numbers a=1/b,0=1/2b, A=1/2b have the required property in Corollary 7.2. In fact, if f(z) satisfies (7.4), then g (z) = bf(z) satisfies the condition mwlg(z)1 ~1, maxlg(z)1 ~b, .:zEE J
zEBz
consequently the values of g (z) cover in generalized sense at least one of the sets S, (j = 1 ,2), which implies that the values of f(z) cover in generalized sense at least one of the sets HoO/b< Iw 1<2/b) and H o (3/b< Iw I <4/b). Theorem 2.9. can be deduced from Corollary 7.2 by applying the latter to the particular case where D is the circle Iz I <1 and E, (j = 1,2) are respectively the point z=O and the circle Izl =1/2.
Extenswns of Some Criterwns
277
Theorem 7. 3. Let D be a domain, Sj 0 = 1,2) two locally uniformly disjoint sets of holomorphic functions in D and p a positive integer. Let
sr be
the
family of the functions f(z) satisfying the following conditions: 1 0 f(z) is hoi om orphic in D. 2 0 T here are two functions
<j:\ (z)
ESt 0 = 1 ,2) such that each of the two func-
tions f(z)-q1(z) 0=1,2) has at most p distinct zeros in D. Then the family
sr is
quasi-nrmal in D of order p at most.
Proof. Consider a suquence of functions fn(z) (n=1,2,···) of the family
sr.
Then to each fn(z) correspond
g. () z =
f. (z) 9lz -
-
911 (z) 911 (z)
has at most p distinct zeros in D and the same is true for gn (z) -1. By Corollary 5.11, the family {gn(Z) (n=1,2,···)} is quasi-normal in D of order p at most. Hence from the sequence gn (z) (n = 1 ,2, ••• ) we can extract a subsequence gn, (z) (k = 1,2, ••• ) converging locally uniformly to a hoi om orphic function or to
00
in a
domain D-a, where a is a set consisting of at most p points of D. Making use of the condition that Si (j = 1,2) are locally uniformly disjoint, we see as in the proof of Theorem 7. 1, that either the sequence fn, (z) (k = 1, 2, ••• ) converges locally uniformly to
00
in D-cr or we can extract from the sequence fn, (z) (k = 1,
2, ••• ) a subsequence f m • (z) (s = 1,2, ••• ) converging locally uniformly to a holomorphic function in D-a. This proves Theorem 7.3.
Definition 7. 4. Let f(z) be a holomorphic function in a doamin D, S a set of holomorphic functions in D and p;?;2 an integer. We say that the values of the function f(z) cover in generalized sense the set S at least p times (strictly at least z times), if for each function qJ(z) E S, the function f(z) -qJ(z) has at least p zeros in D with due count of order of multiplicity (at least p distinct zeros in D).
Corollary 7. 3. Let D be a domain, E t (j = 1,2) two bounded closed sets of points belonging to D, St (j= 1, 2) two locally uniformly disjoint sets of holo-
278
Normal Families of Meromorphic Funct,ons
morphic functions in D, p)!2 an integer and b a positive number. Then there exists a positive number b depending only on D,E"S,(j=1,2), p and b having the following property: If Hz) is a holomorphic function in D such that
I~ 1
If(z)
(7.6)
(j = 1,2,""p)
for p points z j EE 1 (j=1,2,"',p) with
(7.7) and that max If(z)1 )!b, zEEz
then the values of the function Hz) cover in generalized sense strictly at least p times one of the sets S,(j=1,2).
Proof. Assume that the positive number b does not exist. Then for each positive integer n, there is a holomorphic function fn (z) in D such that If.(zJ"))
I~
(7.8)
1 (j = 1,2,···,p)
for p points zf n)EE 1
zi") I)! 0 (j,l
=
1,2,"',p;j
-=F
l),
(7.9)
that max If.(z)
I )! n
(7. 10)
zEEz
and that the values of the function fn (z) do not cover in generalized sense strictly at least p times either of the sets S,(j=1,2). By Theorem 7.3, the family (fn(z) (n=1,2,"')} is quasi-normal in D of order p-l at most. Hence there is a subsequence fn, (z) (k = 1,2, ••• ) converging locally uniformly to a holomorphic func-
Ex ten mons of Some Criterions tion or to
00
279
in a domain D-a where a is a set consisting of at most p-l points of
D. The first case can not happen, because by (7.10) the sequence fn, (z) (k = 1, Z,···) is not uniformly bounded on E 2 • So the sequence fn,(z) (k=l,Z,---) converges locally uniformly to
00
in D-a. by (7. 9) we can find an increasing se-
quence of positive integers k. 0.= 1, Z, ---) such that the p sequences z;m,> 0, = I,Z,---) (j = I,Z,---,p),m), = nt,
converge respectively to the limits /;'jEE 1 (j=I,Z,---,p) with
Is, - sll
~o (j,t = l,Z,---,p ;j=Fl).
(see Lemma 3. Z) One at least of the points /;'j(j=I,Z,---,p), say /;,j,ED-a. This leads to a contradicition in view of (7.8).
Corollary 7_ 4_ Let D, E,(i=I,Z), S,(i=I,Z) and p have the same meaning as in Corollary 7. 3. Then there exists a positive number b depending only on D,E;,SJi=l,Z) and p having the following property: If Hz) is a holomorphic function in D such that If(m)(zo) I::::;:; 1 (m
=
O,l,---,p - 1)
for a point Zo EEl and that
max If(z)1 ~b, %EB z
then the values of the function Hz) cover in generalized sense at least p times one of the sets S;O=l,Z).
Proof - Assume that the positive number b does not exist. Then for each positive integer n, there is a holomorphic function fn (z) in D such that
Normal Families of Meromorphic Functions
280
for a point z~n) EEl' that
and that the values of the function fn (z) do not cover in generalized sense at least p times either of the sets S.(j=1 ,2). Then as in the proof of Corollary 7.3, we find an increasing sequence of positive integers Ix (/. = 1 ,2, ... ) such that the sequence fl, (z) (A= 1,2, ••• ) converges locally uniformly to
00
in D-a, where a is a
set consisting of at most p -1 points of D, and that the sequence Z~I,) (/.= 1,2, ••• ) converges to a limit ~oE E 1 • By Definition 7. 1, we can find a circle
f:
Iz-~o
I ~r belonging to D such that the sets S.(j= 1,2) are uniformly bounded in f and that f -(/;'o)CD-a. Then on the circle Iz-~ol=r the sequence fl,(z) (/.=1,2, ••• ) converges uniformly to
fl,(z) - qi...z)
00,
=
and from the identity
(f1,(Z) -
P).(z)}
+ {P).(z)
- qi...z)},
where ,-I P).(z) =
~ h1,f~)(z~I'»(z
4-0
-
Z~I'»4,
•
we see that, when A is sufficiently large, the function fl, (z) -
r : Iz-~o I
for any
Corollary 7.5. Let D, E.(j=1,2), S,(j=1,2), P and 0 have the same meaning as in Corollary 7. 3. Then there exists a positive number b depending only on D,E" S,(j=1,2), P and 0 having the following property: If Hz) is a holomorphic function in D satisfying the following conditions: 1 0 There are q(2~q~p) points zjEE)(j=1,2,"',q) and q non-negative integers vj(j=1,2,"',q) such that
IZj If(m)(z)
zil I~
~o (j,l
l(m
= 1,2, .. ·,q;j-=/=-l), = 0,1, .. ·,17) (j = 1,2, .. ·,q),
Extens/'ons of Some Critenons
281
g
2: + 1) = j-I (Vj
p;
2°max 1Hz) I?b; zEEz
then the values of the function f(z) cover in generalized sense at least p times one of the sets S,(j=1,2). Remark 7.1. In Corollary 7. 3 and some other similar places, it is tacitly assumed that there exist p points zjEE I (j=1,2,···,p)satisfying the condition (7.
7). Remark 7.2. In the same way that Corollary 7.2 is deduced from Corollary 7. 1, we can deduce respectively from Corollaries 7. 3, 7. 4 and 7. 5 three corollaries on the convering of a set of the form Ho(PI< Iwi
7.2. CASE OF MEROMORPHIC FUNCTIONS
Definition 7.5. Let S,(j=l ,2) be two sets of meromorphic functions in a domain D. We say that S,(j=1,2) are locally uniformly disjoint, if to each point Zo E D correspond a circle IZ-Zo I ,(z)ES,(j=1,2), we have
ICA(Z),CPZ(Z)
I >0
(7. 11)
in the circle Iz-zo I
Theorem 7.4. Let D be a domain and S,(i=1,2,3) three sets of meromorphic functions in D, such that for i#j (i,j=1,2,3) S,and Sjare locally uniformly disjoint and that S,(j=1,2) are normal families in D. Let $r be the family of the functions f(z) satisfying the following conditions: 1 ° f(z) is merom orphic in D. 2° There are three functions q:>,(z) E S, G= 1,2,3) such that each of the equations f(z)=q:>,(z) G=1,2,3) has no root in D. Then the family $r is normal in D. For the proof of this theorem we need the following lemma:
282
Normal Familtes of Meromorp lite Funetwns
Lemma 7. 1. Let fn (z) (n = 1,2, ••• ) be a Co-sequence of meromorphic functions in a domain D and an (z) (n = 1,2, ••• ) be a sequence of holomorphic functions in D converging locally uniformly to a hoi om orphic function in D. Then the sequence of functions gn (z) =fn (z) +a n (z) (n = 1, 2, ••• ) is a Co-sequence in
D.
Proof. Let F (z) be the limit function of the sequence fn (z) (n = 1,2, ••• ) , defined in D (with respect to the spherical distance). Let a(z) be the limit function of the sequence an (z) (n = 1, 2 , ••• ), which is holomorphic in D. Consider a point Zo E D and distinguish two cases: 1) F (zo) #00. Then by Theorem 1. 1, we can find a circle ['0: I Z-Zo I
belonging to D and a positive integer no such that the functions fn (z) (n~no) and F(z) are holomorphic in
['0'
and the sequence fn(z)
(n~no)
converges uniformly
to F (z) in [' o. Accordingly in ['~: IZ-Zo I
lIF (z)
['0'
['0:
Iz-zol
lIfn (z)
(n~no)
and the sequence 1/fn (z) (n~no) converges
in [' o' Writing 1
1
g.(z)
=
1
f.(z) 1
+ _1_
( )'
f.(z)a. z
we see that we can find a circle ['~: IZ-Zo I no)
such that in ['~, lIgn (z) (n~n~) is holomorphic and converges uni-
formly to the function
1 F(z) 1
which is holomorphic in
["0'
1
+
_1_ ( )
F (z)a z
Hence again Zo is a Co-point of the sequence gn (z) (n
= 1,2, ••• ). Since Zo is arbitrary, this proves Lemma 7. 1. N ow let us return to the proof of Theorem 7. 4. Let fn (z) (n = 1 ,2, ••• ) be a
Ex tens/,Ons of Some Cnterions
283
sequence of functions of the family.'iT. Then to each fn (z) correspond three functions an (z) ESp b n(z) ES 2 ,
C n(z)
ES 3 such that each of the three equations
f.(z) = a.(z), f.(z) = b.(z), f.(z) = c.(z)
has no root in D. Since SI (i = 1 ,2) are normal families in D, we can find an increasing sequence of positive integers nk (k = 1, 2, .•• ) such that the two sequences an,. (z), b n, (z) (k = 1,2, ••• ) are Co-sequences in D and hence have respectively limit functions a (z), b (z) defined in D with respect to the spherical distance. For each point zED, the two corresponding values a (z), b (z) must be unequal. In fact, if at a point zoED we have a(zo)=b(zo), then the inequality
yields lan(zo),bn(zo) I~O (n~+oo). This contradicts the inequality lan(zo),b n (zo) I>0 resulting from (7. 11). N ow consider a point Zo E D. Then one at least of the values a (zo) and b (zo)
is finite, say a (zo):;6oo. By Theorem 1. 1, we can find a circle ro: IZ-Zo I
(7.12)
and Ib •• (z),a •• (z)1 >0, Ic •• (z),a •• (z)
I >0,
Ib •• (z), c•• (z)
where 0<0<1 is independent of k. Consider the three sequences
F.(z)
=
.
1
f. (z) -
·
a. (z)
I >0,
(7.13)
Normal Families of Meromorphic Functions
284
B.(z)
=
1
b ( ) _ at
Z
a ...
( ) ' C.(z) Z
=
1
Cat
() _ Z
a. t
(). Z
Fk(z) ,Bk(z) ,Ck(z) are holomorphic in ro and each of the two equations F. (z) = B. (z), F. (z) = C. (z)
has no root in rD. Moreover by (7.13), we have
in r o , and by (7.12),(7.13) and Lemma 1.1, we have
IB.(z) -
C.(z)
I~
IB.(z) ,C.(z)
1
1
I = I B.(z) 'C.(z) I 1
1
Ib.,(z) -a.,(z), c"(z) -a.,(z)1 ~21 +M2o
in rD. Consequently by Theorem 7.1, the family Fk(z) (k~ko) is normal in r o, hence the family
1
F.(z) = f., (z) -
a., (z) (k ~ k o)
is normal in rD. Then by Lemma 7.1, we see that the identity f., (z) =
{f., (z) - a., (z)}
+ a" (z)
(k ~ k o)
implies that the family fn, (z) (k~ko) is normal in rD. Again this implies that the family fn, (z)
(k~l)
is normal in rD.
Since Zo is arbitrary, the family fn, (z) (k ~1) is normal in D, by Theorem 1. 4. Hence we can extract from the sequence fn, (z) (k~l) a subsequence fm, (z) (h~l)
which is a Co-sequence in D. This proves Theorem 7.4.
Definition 7. 6. Let f(z) be a meromorphic function in a domain D and S
285
Extenswns of Some Criterions
a set of meromorphic functions in D. We say that the values of the function Hz) cover in generalized sense the set S, if for each function qJ(z)ES, the equation f (z)=qJ(z) has at least one root in D.
Definition 7. 7. Let E be a bounded closed set of points of the complex plane C. Consider a system of a finite number of circles
r:
Iz-zjl
... ,m) such that
..
E CU(lz -
zjl
j-1
The linear measure of E, denoted by AE, is defined by
. AE
= tnf(L;rj)' r
j-1
where the inf is taken over all such systems
r
of circles.
As an example, consider a continuous curve C: z=z(t) (a~t~b), where z (t) is a continuous function in the interval a~t~b such that z(a)#z(b). It can be shown that
AC ~
1 21 Z (a)
-
Z
(b)
I.
Corollary 7.6. Let D be a domain, E,(j=1,2) two bounded closed sets of points belonging to D, S,(j=1,2,3) three sets of meromorphic functions in D such that S,and Sjare locally uniformly disjoint for i#j (j,j=1,2,3) and that S, (j=1,2) are normal families in D, and
t
a positive number. Then there exists a
positive number b depending only on D, E,(j=1,2), S,0=1,2,3) and
t,
having
the following property: If Hz) is a meromorphic function in D such that mmlf(z)I~1,A(zlzEE2' If(z)l~b)~e, zEEl
(7. 14)
286
Normal Families of Meromorphic Functions
then the values of the function f(z) cover in generalized sense at least one of the sets SI(i=1,2,3).
Proof. Assume that the positive number b does not exist. Then for each positive integer n, there is a meromorphic function fn (z) in D such that m~n If. (z) I ~ 1, A (z Iz zEE J
E E 2' If. (z) I ~ n)
~ e
(7. 15)
and that the values of the function fn (z) do not cover in generalized sense either of the sets SI(i=1,2,3). By Theorme 7.4, the family {fn(z) (n=I,2, .. ·)} is normal in D. Accordingly we can find a subsequence f.. (z) (k = 1,2, ... ) which is a Co-sequence in D. Let F(z) be the limit function of the sequence fn, (z) (k = 1, 2,,,,), defined in D with respect to the spherical distance. In view of (7.15), F (z) can not be the constant orem 1. 3. Let
r;:
00,
so F (z) is a meromorphic function in D, by The-
~j(j=I,2, .. ·,q)
be the poles (if there is any) of F(z) in E 2, and
.
IZ-~j I
(j = 1,2, ... , q) are mutually disjoint and interior to D. The set E =E2 -
-
rj
• r j is a U
j-I
bounded closed set of points of D. Since
there is a point
Zk
E E such that (7. 16)
On the other hand, since F(z) has no pole on E, it is bounded on E: IF(z)I~A (O
Take a number B>A and let 00,
have
~=~(A
,B) defined in Lemma 1. 2. Since as k---1+
Ifn,(z), F(z) 1---10 uniformly on E, there is kosuch that when k~ko' we
287
Extens1,Ons of Some Gntenons If •• (z),F(z)
1< Jl
fOT
Z
E E,
which implies
If •• (z)
1< B
fOT
Z
E
E.
This contradicts (7.16). Note that in Corollary 7.6, the first inequality in (7.14) is equivalent to
1
max If(z),ool ~--
Vz
'EE,
and the second inequality is equivalent to
This suggests us to generalize the condition (7. 4) by replacing the point an arbitrary point
Wo
00
by
E C. In this way we get the following corollary:
Corollary 7.7. Let D, E i (j=I,Z), Si(j=l,Z,3) and meaning as in Corollary 7.6. Let
Wo
be a point of
C and
b
f
have the same
(O
a number.
T hen there exis ts a number 11 (0<11< 1) depending only on D, Ei (j = 1 , Z), Si (j =
1 , Z , 3) , f,
Wo
and b, having the following property: If f (z) is a meromorphic
function in D such that max If(z),wol ~o, A(zlz zEE J
E
E 2 , If(z),wol <'TJ) ~e,
(7.17)
then the values of the function Hz) cover in generalized sense at least one of the sets S,(j=l,Z,3). For the proof of this corollary, we need the following lemma:
Lemma 7. 2. Let f(z) be a meromorphic function in a domain D and point of
C.
Then the function 1Hz),
Wo
I of z is
continuous in D.
Wo
a
Normal Famihes of Meromorphtc Functions
288
Proof. Consider a point zoED. We have If(z),wol ~ If(z),f(zo)1 If(zo),wol ~ If(zo),f(z)1
+ If(zo),wol, + If(z),wol,
hence Ilf(z),wol- If(zo),woll ~ If(z),f(zo)l,
which imples that the function If(z),wolis continuous at zo, by Lemma}. 4. Now let us come back to the proof of Corollary 7.7. Assume that the number 11(0<11<1) does not exist. Then for each positive integer n, there is a meromorphic function fn (z) in D such that
max If.(z) ,Wo I ~ 6, A(z Iz E E 2 , If.(z) ,Wo I ~ tEE l
1 -+ 1) ~ e
n
(7.18)
and that the values of the function fn (z) do not cover in generalized sense either of the sets SI(i=1,2,3). By Theorme 7.4, the family {fn(z) (n=I,2,···)} is normal in D. Accordingly we can find a subsequence fn, (z) (k = 1,2, ••• ) which is a Co-sequence in D. Let F (z) be the limit function of the sequence fn, (z) (k = 1, 2, ••• ), defined in D with respect to the spherical distance. In view of the first inequality in (7.18), F(z) can not be the constant woo Let ~J(j=1,2,···,q) be the roots (if there is any) of the equation F(z)=wo in E 2 , and
r j:
Iz-~d
q
2,···,q) be circles such that ~rJ<£ and the closures Tj (j=I,2,···,q) are mutuJ-1
q
ally disjoint and interior to D. T he set E = E2 -
U r J is a bounded closed set of
j-1
points of D. Since
1 -+ 1) ~ e,
A(z Iz E E 2 , If.• (z) ,Wo I ~ n.
there is a point z. E E such that
Ex tenmons of Some Criterions 1 -+ l'
If. • (z.),wol ~ n.
289
(7. 19)
On the other hand, by Lemma 7.2, the function IF(z), wol is continuous in D, hence
IF(z),wol ~m for z E E, where m>O is a constant. Next from the inequality
IF(z),wol ~ IF(z),f •• (z)1
+ If •• (z),wol
and the fact that as k-j+=, IF(z), fn,(z) I-jO uniformly on E, we see that we can find ko such that for k~ko, we have
m If •• (z),wol ~2 for z E E.
This contradicts (7.19). Remark 7. 3. In Corollaries 7. 6, 7. 7 and in other similar places, it is tacitly assumed that AE2~E. Instead of assuming the function Hz) satisfying a condition of variation of modulus of the form (7.14) in Corollary 7. 6 or a condition of variation of spherical distance of the form (7.17) in Corollary 7.7, we can also assume the function Hz) to satisfy a condition involving the spherical derivative C1(z,f). (see O.
28»
Corollary 7. 8. Let D be a domain, E a bounded closed set of points belonging to D and S, (j = 1 ,2,3) three sets of meromorphic functions in D such that S, and Sj are locally uniformly disjoint for i#j (j ,j= 1,2,3) and that S,(j= 1, 2) are normal families in D. Then there exists a positive number 13 depending only on D,E and S,(j= 1,2,3) having the following property: If Hz) is a meromorphic function in D such that
maxC1(z,f) .EB
~p,
(7. 19)
290
Normal Families of Meromorphic Functions
then the values of the funciton f(z) cover in generalized sense at least one of the sets S,Ci=1,2,3).
Proof. Assume that the positive number 13 does not exist. Then for each positive integer n, there is a meromorphic function fn (z) in D such that maxa(z
,f.)
~ n
(7.20)
zEE
and that the values of the function fn (z) do not cover in generalized sense either of the sets S,Ci=1,2,3). By Theorem 7.4, the family {fn(z) (n=1,2, .. ·)} is normal in D. Consequently by Theorem 1. 6, the sequence of functions a(z,fn ) (n = 1 ,2,,,,) is locally uniformly bounded in D, and hence it is uniformly bounded on E. This contradicts (7.20). We can also assume the function fCz) to satisfy a condition involving the spherical oscillation 0 (zo,r,f) defined as follows:
Definition 7. 8. Let fez) be a meromorphic function in a domain D and c:
IZ-Zo I
a circle belonging to D. We define O(zo,r,f) =
sup If(z]),f(z2)1 %l'%2 Ec
which is called the spherical oscillation of the function f(z) in c.
Definition 7. 9. Consider a domain D and a bounded closed set E of points belonging to D. It is known that we can find a positive number p such that for each point Zo E E, the circle
Iz -Zo I
to D. We denote by R (D, E)
the least upper bound of the set of such positive nubmers p.
Corollary 7. 9. Let D,E and S,Ci=1,2,3) have the same meaning as in Corollary 7.8. Let 0<£<1 be a number. Then there exists a number O
Ex ten S/,ons of Some Cnterions
291 (7. 21)
then the values of the function f(z) cover in generalized sense at least one of the sets S I O=1,2,3).
Proof. Assume that the number r does not exist. Take a number O
(7. 22)
and that the values of the function fn(z) do not cover in generalized sense either of the sets S j O=1,2,3). By Theorem 7.4, the family (fn(z) (n=1,2,"')} is normal in D. By Theorem 1. 5, the family {fn(z) (n = 1, 2, .. ·)} is equicontinuous in D with respect to the spherical distance. Since E is bounded closed, we can find a finite number of points 1;,j EE (j=1,2,"',m) such that
Consider the bounded closed set m
r=U(lz-sjl j~1
~p)CD.
By the equicontinuity of the family (In (z) (n = 1 , 2, ... ) }, corresponding to the positive number e/2, there is a number &>0 such that
N ext take a positive integer no>2 such that
Normal Families of Meromorphic Functions
292
Consider a point zoE E. Then evidently
and hence for any two points ZI ,Z2 of the circle
IZ-Zo I
we have
and therefore
Since Zo E E is arbitrary, we have
which contradicts (7.22).
Theorem 7. 5. Let D be a domain, S, 0 = 1, 2,3) three sets of meromorphic functions in D, such that for i#j 0 ,j= 1,2,3) Sj and Sj are locally uniformly disjoint and that S,0=1,2) are normal families in D, and p)':l an integer. Let .§T
be the family of the functions f(z) satisfying the following conditions: 1 0 f(z) is merom orphic in D. 2 0 T here are three functions
q< (z)
E S, 0 = 1 ,2,3) such that each of the equa-
tions f(z)=q«z) 0=1,2,3) has at most p distinct roots in D. Then the family .§T is quasi-normal in D of order p at most. For the proof of this theorem, we need some lemmas.
Lemma 7. 3. Let fn (z) (n = 1,2,,,,) be a sequence of merom orphic functions in a domain D,
qJ.
(z), 1Vn (z) (n = 1,2, ... ) two sequences of holomorphic
functions in D and p)': 1 an integer. Suppose that the following conditions are satisfied:
293
Ex tenmons of Some Cnterions
1° For each n. each of the equations fn(z)=oo. fn(z)=qJ,,(z). f n(z)=1I>.(z) has at most p distinct roots in D. 2° For each point zoED. we can find a circle Iz-zol
¢. (z) I > <5 (n = 1.2 ... ·)
in the circle IZ-Zo I
Proof. By the condition 2° in Lemma 7. 3. the two sequences
qJ" (z).
1I>n (z)
(n = 1, 2 .... ) are locally uniformly bounded in D. hence by Corollary 1. 3. we can find an increasing sequence of positive integers nt (k = 1. 2 ... ·) such that the two sequences <:A.. (z).
11>.. (z)
(k = 1. 2 ... ·) converge locally uniformly to holomor-
phic functions (z) respectively in D. Evidently by 2°. we have I
¢(z) I ~ <5 for Iz -
zol < r.
(7.23)
Now consider the sequence of functions
meromorphic in D. By the condition 1° in Lemma 7. 3. for k ~ 1. each of the three equations F.(z) = O. F.(z) = 1, F.(z) =
00
has at most p distinct roots in D. consequently by Corollary 5.9. the family {F t (z) (k = 1.2 ... ·)} is quasi-normal in D of order p at most. Accordingly there is a subsequence Ft.(z) (s=I.2 ... ·) which is a Co-sequence in a domain D-oo. where 00
consists of at most p points of D. Evidently this is also true for the sequence F •• (z) {1/J ... (z) -
294
Normal Famtlws of Meromorphic Functions
by Theorem 1. 1 and (7.23). Then by Lemma 7.1, we see that the identity
implies that the sequence f m • (z) (s = 1,2,,") is a Co-sequence in D-w. Evidently, in general, from every subsequence fA,. (z) (k = 1,2,"') of the sequence fn (z) (n = 1,2, ... ), we can extract a subsequence f •• (z) (s = 1,2, ... ) which is a Co-sequence in a domain D-a, where a consists of at most p points of D. Therefore the family {fn(z) (n = 1,2, .. ·)} is quasi-normal in D of order p at most.
Lemma 7. 4. Let S: fn (z) (n = 1,2, ... ) be a sequence of meromorphic functions in a circle IZ-Zo I
(7. 24)
has at least one root in the circle Iz-zol
Proof. Distinguish several cases: 1) f(z) and a(z) merom orphic functions.
Assume first that a(zo):;eoo. Since f(z)-a(z)~O, there is a circle c: Iz-zo I =p (O
1t--1+OO
and
=
f(z), ltm a.(z) 1t--1+ OO
=
a(z) untformly on c
295
Extensions of Some Cntenons
If (z)
a (z) I
-
?
m
>
0 on c.
Now suppose that there does not exist a positive integer N having the required property in Lemma 7. 4. Then we can find an increasing sequence of positive integers n. (k = 1, 2, .•. ) such that for k? 1, the equation
f •• (z) =
a •• (z)
has no root in the circle Iz-zol
F.(z)
= f •.c z )
1
_ a •• (z) (k
=
1,2,,,,)
are holomorphic in the circle IZ-Zo I
11 (fCz) -a (z) },
it is uniformly convergent in the circle IZ-Zo I <po This implies
that the point Zo is a Co-point of the sequence
1
) F1 ( z
= f. • (z)
-
a. (z) (k !
= 1,2,,,,).
Now since a(zo)#=, by Theorem 1. 1, we can find a circle
r,
r : Iz-zo I
an" (z) (k?k o) and a(z) are holo-
morphic and the sequence an" (z) (k ?k o) converges uniformly to a (z). Then by Lemma 7. 1, the identity
f. • (z) =
1
F.. -z()
+ a.• (z)
implies that the point Zo is a Co-point of the sequence fn" (z) (k = 1,2,,,,), This contradicts the hypothesis that the sequence S is irreducible with respect to Z00 Next assume that a (zo) ==. Then two cases are possible: A) fCz) ~O. In this case it is sufficient to apply the above result to lIfn (z) and l/a n (z) (n?no). B) fCz)=O. In this case we take a constant 13#0 and apply the result in case A) to f n (z)+13, a n (z)+13 (n?!). 2) fCz) is the constant = and a (z) is a meromorphic function. Then two
296
Normal Families of Meromorphtc Functions
cases are possible: AI) a(z)~O. In this case we apply the result in case 1) to l/fn(z) and l/a n (z)
(n~no).
B I ) a(z)=O In this case we take a constant 13*0 and apply the result in case AI) to f n(z)+I3, a n(z)+13
(n~1).
3) fez) is a merom orphic function and a (z) is the constant =. Then again
we distinguish two cases: A z) f(z)~O. In this case we apply the result in case 1) to l/f.(z) and l/a. (z)
(n~no).
B z ) f(z)=O. In this case we take a constant 13*0 and apply the result in case A z) to fn(z)+I3, a n(z)+13 (n~1). The proof of Lemma 7.4 is now complete. Now let us prove Theorem 7.5. Let fn(z) (n=1,2,···) be a sequence of functions of the family.ffr. By hypothesis, to each fn (z) correspond three functions an (z) E SI' b n(z) E Sz, Cn(z) ES3 such that each of the three equations f.(z) = a.(z), f.(z) = b.(z), f.(z) = c.(z)
has at most p distinct roots in D. Since S, (j = 1,2) are normal families in D, there is an increasing sequence of positive integers n. (k = 1,2, ••• ) such that the two sequences an, (z), b n, (z) (k = 1,2, ••• ) are Co-sequences in D. Let a(z), b (z) be respectively the limit functions defined in D with respect to the spherical distance, ofan,(z), b.,(z) (k=1,2,···). Consider a point zoED. Then one at least of the values a(zo) and b(zo) is finite, say a (zo) *=. By Theorem 1. 1, we can find a circle r
0:
Iz -Zo I
longing to D and a p~sitive integer ko such that the functions an, (z) (k~ko) and a (z) are holomorphic in ro and the sequence an, (z) (k~ko) converges uniformly to a(z) in
roo
Then as in the proof of Theorem 7. 4, we introduce the sequence of meromorphic functions in r
0:
1 F.(z) = f., (z) _ a., (z) (k ~ k o)
and the two sequences of holomorphic functions in r
0:
297
Extenmons of Some Criterions
B.(z)
=
1
b ( ) _ "a
z
a "i
( ) ' C.(z) Z
=
1
() _
c.. z
a "i
() Z
(k ~ k o).
We may assume that ro is small enough, so that in rowe have
IB.(z)
1 1< 6'
Ic.(z)
1 1< 6'
IB.(z) -
C.(z)
I> 1/,
where band 11 are positive numbers. Since each of the three equations F.(z) =
00,
F.(z) = B.(z), F.(z) = C.(z)
has at most p distinct roots in r o, we can apply Lemma 7. 3 and conclude that the family {Ft(z) (k~ko)} is quasi-normal in ro of order p at most. This implies that the family
1
) F1 ( Z
= f.
(z) -
a. (z) (k ~ko)
)I
i
is also quasi-normal in ro of order p at most. Then by Lemma 7.1, the identity
f. .t (z) =
1
-I F (Z )
+ a. (z) .t
implies that the family {fn, (z) (k~ko)} is quasi-normal in ro of order p at most. Of course the same is true for the family {in, (z) (k = 1, 2, ••• ) }. Since Zo E D is arbitrary, we conclude, by Theorem 5.1, that the family {fn, (z) (k = 1,2, ••• )} is quasi-normal in D. Accordingly we can extract from fn, (z) (k = 1 ,2, ••• ) a subsequence f m• (z) (h =
1,2, ••• ) which is a C)-sequence in D. Then by Lemma 5. 5, we can extract
from f m • (z) (h = 1,2, ••• ) a subsequence fo, (z) (j = 1,2, ••• ) which is either a C osequence in D or an irreducible C)-sequence in D. Consider the latter case and let E be the set of the nonCo-points of the sequence fo,(z) (j=1,2,···) in D. Let f (z) be the limit function defined in D-E with respect to the spherical distance, of the sequence f.,(z) (j=1,2,···). Consider the seqeunces ao,(z), b.,(z) (j=1,2, ••• ) and their limit functions a (z), b (z) defined above. One at least of the two
298
Normal Families of Meromorp hic Functions
functions a(z).b(z) say a(z) is such that a(z)~f(z) in D-E. Assume that there are v points z,(j=1,2 ... • .v) belonging to E. Let
r ,: Iz-z, l
be
r, containing
no
v circles belonging to D. without common point each other and
point of E except z,. Consider a circle
r ,.
Since f(z)~a(z) in the domain 0< Iz-
z,1
has at least one root in the circle
r ,.
Let J=maxJ,. then when j~J. the equation l~i
(7. 25) has at least one root in each of the circles
r, 0 =
1. 2 ..... v). It follows
that ~p and E consists of at most p points. Since f" (z) (j= 1. 2 ... ·) is a subsequence of the sequence fn(z) (n=I.2 ... ·). we have proved that the family 57 is quasi-normal in D of order p at most.
Definition 7. 10. Let f(z) be a meromorphic function in a domain D and S a set of merom orphic functions in D and p~2 an integer. We say that the values of the funciton f(z) cover in generalized sense the set S strictly at least p times. if for each function qJ(z) E S. the equation f(z) =qJ(z) has at least p distinct roots in D.
Corollary 7. 10. Let D be a domain. E, (j= 1. 2) two bounded closed sets of points belonging to D. S,0=1,2.3) three sets of meromorphic functions in D such that S, and Sj are locally uniformly disjoint for i*j O.j = 1.2.3) and that S, 0= 1. 2) are normal families in D.
E
and d positive numbers and p~2 an integer.
Then there exists a positive number b depending only on D.E,0=1,2). S,O=l. 2.3). E.d and p. having the following property: If Hz) is a meromorphic function in D satisfying the following conditions: 1° If(zj) 1~1 for p points zjEE j (j=1,2 ... ·.p) with IZj-z.l~d (j.k=I.2 • .. ·.p;j*k); 2° A (z IzE E 2 • If(z) I ~b)~E; then the values of the function Hz) cover in generalized sense at least one of the sets S,0=1.2.3) strictly at least p times.
299
Ex ten $l,onS of Some Criterions
Proof. Assume that the positive number b does not exist. Then for each positive integer n, there is a meromorphic function fn (z) in D satisfying the following conditions: 1) Ifn (zJn» I ~1 for p points Zj(n) E EI (j= 1,2, ••• ,p) with IZj(n)_z~n) I ~d (j,k
=1,2,···,p;j#k). 2)A(zlzEE z , Ifn(z) I~n)~e. 3) The values of the function fn (z) do not cover in generalized sense either of the sets S;Ci=I,2,3) strictly at least p times. By Theorem 7. 5, the family {fn (z) (n = 1, 2, •.• )} is quasi-normal in D of order p -1 at most. Accordingly from the sequence fn (z) (n = 1,2, ••• ) we can extract a subsequence fn,. (z) (k = 1,2, ••• ) which is a Co-sequence in a domain D()), where ()) is a set consisting of at most p-l points of D. Let Hz) be its limit function defined in D-()), with respect to the spherical distance. Hz) can not be the constant
00. In fact, we can find an increasing sequence of positive integers
k, (s = 1, 2, ••• ) such that the p sequences zJ".)(S= 1,2,···) (j
= 1,2,···,p),m. = n ••
converge respectively to limits <;'j EEl (j = 1 ,2, ••• , p) with
(see Lemma 3. 2) Among the points <;,; (j = 1 ,2, ••• , p) , there is one <;,;. E DI -()). If Hz) =00, then in a circle Iz-<;'j. I <6, we have uniformly
lim If .. (z),ool = 0,
.....-.)+00
•
which is incompatible with the fact:
So Hz) is a meromorphic function in D-()). Let a,(j= 1,2, ••• ,q) be the points of the set Ez
n()).
Describe circles
r;:
q
Iz-a, I
Normal Famtlies of Meromorphic Functwns
300 Q
set a=E 2 -
UrjCD-w is bounded closed. Next let !3j(j=1,2,"',q') be the j-1
poles of the function Hz) on a. Again describe circles
r; : Iz-!3j I
Q'
q') such that j~/;
tually disjoint. Then a. =a- U
j-1
r; CD-w is
a bounded closed set on which Hz)
is finite. By the condition 2), for each k there is a point ak Ea. such that If •• (a.)
I ~n ••
On the other hand, since Hz) is finite on
a.,
(7.26)
by Lemma 5.13, when
k~ko,
we
have If •• (z)
I<
B for z
E
(T. ,
where B is a positive constant independant of k. This contradicts (7.26).
Corollary 7.11. Let D, E t (i=1,2), St(i=1,2,3), e,d and p have the same meaning as in Corollary 7. 10. Let Wo be a point of
C and
1\(0<1\<1) a
number. Then there exists a number 11 (0<11<1) depending only on D, E j(j = 1, 2), Sj(j=1,2,3), e,d,p, woand 1\, having the following property: If Hz) is a merom orphic function in D satisfying the following conditions: 1 0 IHzj) ,Wo I ~1\ for p points zjE E1 (j= 1,2, ••• ,p) with IZj-Zk I ~ (j,k = 1, 2,"',p;j#k); 2 0 A(zlzEE 2 , IHz),wol~11)~e; then the values of the function Hz) cover in generalized sense at least one of the sets St(i=1,2,3) strictly at least p times. Just as the proof of Corollary 7.7 is quite similar to that of Corollary 7.6, the proof of Corollary 7. 11 is quite similar to that of Corallary 7.10. However for the sake of completeness, we give briefly the proof.
Proof. Assume that the number 11(0<11<1) does not exist. Then for each positive integer n, there is a merom orphic function fn (z) in D satisfying the following conditions:
Ex ten s'tOns of Some Criterions
301
1) Ifn (zJn) , Wo I ~o for p points zin) EEl (j = 1 ,2, ••• , p) with Izin) , z~n) I ~d (j,k =1,2, ••• ,p ;j:;;6k).
1
2) A(zlzEE 2 , Ifn(z),wol~n+l)~e. 3) The values of the function fn (z) do not cover in generalized sense either of the sets SiO=I,2,3) strictly at least p times. By Theoerm 7. 5, the family {fn(z) (n = 1,2,"')} is quasi-normal in D of order p -1 at most. Accordingly from the sequence fn (z) (n = 1,2,,,,) we can extract a subsequence f .. (z) (k=1,2, .. ·) which is a Co-sequence in a domain D-w, where
w is a set consisting of at most p-l points of D. Let f(z) be its limit function defined in D-w, with respect to the spherical distance. By the condition 1), we see as in the proof of Corollary 7. 10, that Hz) is not the constant woo Having obtained this result, the proof can then be completed by a combination of the method used in the proof of Corollary 7. 10 and that used in the proof of Corollary 7.7.
Corollary 7. 12. Let D be a domain, E a bounded closed set of points belonging to D, S,0=1,2,3) three sets of meromorphic functions in D such that S, and Sj are locally uniformly disjoint for i:;;6j (j,j = 1,2,3) and Si 0 = 1,2) are normal families in D, d a positivs number and p~2 an integer. Then there exists a positive number 13 depending only on D, E, S, 0 = 1, 2,3), d and p, having the following property: If f(z) is a merom orphic function in D such that a(z j , f) ~
P (j =
1,2,,,,, p )
for p points zjEE (j=1,2,"',p) with IZj-zkl~d (j,k=1,2,"',p;j:;;6k), then the values of the function f(z) cover in generalized sense at least one of the sets Si 0= 1,2,3) strictly at least p times.
Proof. Assume that the positive number 13 does not exist. Then for each positive integer n, there is a meromorphic function fn(z) in D such that
302
Normal Families of Meromorphzc Functwns
and that the values of the function fn (z) do not cover in generalized sense either of the sets S,(j=1,2,3) strictly at least p times. By Theorem 7.5, the family {fn (z) (n = 1,2, ••• )} is quasi-normal in D of order p -1 at most. From the sequence fn (z) (n = 1,2, ••• ) we can extract a subsequence fn, (z) (k = 1,2, ••• ) which is a Co -sequence in a domain D-w, where w is a set consisting of at most p-1 points of D.
A fortiori,
the family {fn, (z) (k = 1 , 2, ••• )} is normal in D-w, and by T he-
orem 1. 6, the sequence of functions a(z, f n,) (k = 1 , 2, ••• ) is locally uniformly bounded in D-w. On the other hand, let k, (s = 1 , 2, ••• ) be an increasing sequence of positive integers such that the p sequences Zj(m,) (s = 1,2",,) (j = 1,2,,,,, p) , m,=n.,' converge respectively to limits /;,jEE (j=1,2,"',p) with I/;,j-/;,.I~d (j,k=1,2,"·,p;j#k). (see Lemma 3.2) One at least of the points /;,j(j=1,2, ... , p), say /;,j, E D-w. Consequently the sequence a(z, fn,) (k = 1,2, ... ) is uniformly bounded in a circle Iz-/;,jo I<po This is incompatible with the fact: a(z(m'),f ) Jo
m.
~
m , lim z(m,) II
H+OO
10
= S. )0
Corollary 7. 13. Let D, E, S,(j=1,2,3), d and p have the same meaning as in Corollary 7. 12. Let O<E
~e
(j = 1,2, .. ·,p)
for p points zjEE (j=1,2,''',p) with IZj-z.l~d (j,k=1,2,"',p;j#k), then the values of the function f(z) cover in generalized sense at least one of the sets S,(j=1,2,3) strictly at least p tiems.
Proof. Assume that the number r does not exist. Take a number O
(7. 27)
Extensions of Some Criterwns
303
for p points zj(n)EE (j=I,2,"',p) with Iz{n)-z~n)l~d (j,k=I,2,"',p;j#k) and that the values of the function fn (z) do not cover in generalized sense either of the sets S,(j= 1,2,3) strictly at least p times. By Theorem 7.5, the family {fn (z) (n=1 ,2,"')} is quasi-normal in D of order p-l at most. From the sequence fn (z) (n = 1,2, ••• ) we can extract a subsequence f .. (z) (k = 1,2, ••• ) which is a Co-sequence in a domain D-ro, where ro is a set consisting of at most p -1 points of D.
A fortiori,
the family {f.. (z) (k = 1,2, ••• )} is normal in D-ro, and hence,
by Theorem 1. 5, the family
if.. (z)
(k = 1 ,2, ••• )} is equicontinuous in D-ro with
respect to the spherical distance. Let k. (s = 1,2, ••• ) be an increasing sequence of positive integers such that the p sequences Zj(m,) (s = 1,2,,,,) (j = 1,2,'" , p), m. =n., converge vespectively to limits I;.jEE (j=1,2,"',p) with Il;.j-I;..I~d (j,k = 1,2,·",p;j#k). One at least of the points I;.j(j=1,2,"·,p), say I;.j,ED-ro. Since the family ifn, (z) (k = 1 ,2, ... )} is equicontinuous at the point I;.j,' we can find a circle c: Iz-l;.j,1
Take a sufficiently large integer s such that
Then the circle Iz-z}m,) , I<.£... m, belongs to c, and hence
which is incompatible with (7.27).
7.3. CASE OF UNIVALENT AND MULTIVALENT FUNCTIONS In this part the following lemma plays an important role:
Lemma 7.5. Let.'iJT be a normal family of holomorphic functions f(z) in a domain D satisfying the following conditions:
Normal Families of Meromorphic Functions
304
1 0 The functions f(z) of the family .7 have no zero in D.
2 0 There does not exist a sequence of functions fn (z) (n = 1. 2 ••.• ) of the family .7. which converges locally uniformly to zero in D. Let E be a bounded closed set of points belonging to D. Then there exists a number A>O such that for each function f(z) E.7 we have If(z)1 ~),for z E E.
(7.28)
Proof. Assume that such a number A does not exist. Then to each positive integer n corresponds a function fn (z) E.7 such that
min If.
I < 1... n
(7.29)
From the sequence fn (z) (n = 1. 2 •••• ) we can extract a subsequence fn, (z) (k = 1.
2 •••• ) converging locally uniformly to a holomorphic function g (z) in D. By hypothesis g(z):;i:O. From (7.29) evidently g(z) has at least one zero zoEE. Then by a well known theorem. when k is sufficiently large. fn, (z) has zero in a certain circle with center zoo So we get a contradiction.
Theorem 7. 6. Let D be a domain and E). E2 two bounded closed sets of points belonging to D. Then there exists a number p> 0 depending only on D. E).E 2 having the following property:1f f(z) is a holomorphic and univalent function in D satisfying the following conditions: 1 0 Hz) has a zero zoEE);
20 Ifl
(z~) I ~1 at a point z~E E 2;
then the values of f(z) cover in generalized sense the set Ho( Iw I
Proof. Denote by .7 the family of the functions Hz) satisfying the conditions in Theorem 7. 6. Assume that there does not exist a number p> 0 having the required property. Then to each positive integer n corresponds a function fn (z) E.7 such that the values of fn (z) do not cover in generalized sense the set Ho ( Iwi <
1..). n
Let Zn E E) be the zero of fn (z). We can find a subsequence
~. =Zn
'
(k
Ex ten 81,ons of Some Criterions
305
=1,2,,") converging to a point t;,oEEI" Consider a circle Iz-t;,ol~r belonging to D and the domain
Do = D - (Iz - ~ol ~r!2). Let ko be a positivs integer such that, for k~ko' t;,. lies in the circle Iz-t;,o I O such that for each k~ko we have Ig.(z)l~pfor Iz-~ol =r.
Now consider an integer k~koand a function cp(z)EH o( Iw I
n.
Theorem 7. 7. Let D be a domain, EIJ E2 two bounded closed sets of points belonging to D and p~2 an integer. Then there exists a number p>O depending only on D,E IJ E 2 and p, having the following property: If Hz) is a holomorphic and p-valent function in D satisfying the following conditions: 1 0 Hz) has a zero Zo E E1 of order p; 2°lf(p)(z~) I~l
at a point
z~EE2;
then the values of f(z) cover in generalized sense the set Ho( Iwi
306
Normal Families of Meromorphtc Functions
Theorem 7. 8. Let D be a domain and E, (j = 1, 2,3) three bounded closed sets of points belonging to D, such that (7.30) T hen there exists a number p> 0 depending only on D and E, (j = 1,2,3) having the following property: If f(z) is a merom orphic and univalent function in D satisfying the following conditions: 1 ° f(z) has a zero zoE E l ; 2 0 1ft (z~)I~l at a point z~EE2; 3° f(z) has a pole z~ E Eli then the values of f(z) cover in generalized sense the set Ho( Iwi
Proof. Denote by
$T the family of the functions f(z) satisfying the condi-
tions in Theorem 7.8. Assume that there does not exist a number p>O having the required property. Then to each positive integer n corresponds a function fn (z) E $T such that the values of fn (z) do not cover in generalized sense the set Ho
( Iwi < ~). n
Let Zn E E3 be the pole of fn(z). We can find a subsequence /;,. =Zn (k '
= 1, 2, ... ) converging to a point /;,0 E E 3. Consider a circle
r: Iz-/;'o I~r belong-
ing to D and such that El and E2 are exterior to r. Such a circle r can be found by (7. 30). Let k 0 be a positive integer such that for k ~k 0,
/;,.
lies in the circle r :
I z-/;'o I0 such that, for k ~k 0' the values of g. (z) cover in generalized sense the set Ho. ( Iw I
~),
n.
if l/n.
Theorem 7.9. Let D be a domain, E,(i=1,2,3) three bounded closed sets of points belonging to D satisfying the condition (7. 30) and
p~2
an inte-
ger. T hen there exists a number p> 0 depending only on D, E, (j = 1,2,3) and p, having the following property: If f(z) is a meromorphic and p-valent function in D satisfying the following conditions:
307
Extenmons of Some Criterions 1° f{z) has a zero zoE El of order p; 2°lf(p)(z~) I~l at a point z~EE2;
3° f{z) has a pole z~E E3 of order p; then the values of f(z) cover in generalized sense the set HD ( Iw I
m~O
is an integer. Then p will depend also on m. The same remark is al-
so applicable to Theorems 7.8 and 7.9 in the sense that any condition C which can be used instead of the condition 2° in Theorems 7.6 and 7.7, can also be used instead of the condition 2° in Theorems 7.8 and 7.9. Remar k 7. 5. In Theorems 7. 7 and 7. 9, the condition 1 ° may be replaced by "f{z) has
k(k~2)
zeros a j EE 1 (j=1,2,"',k) of orders a,(j=1,2,"',k) respec-
k
tively with ~ a,=p and laj-at I ~o O:::;;;;j,j' :::;;;;k ,j#i' ;0>0). " Also in Theorem j~l
7.9, the condition 3° may be replaced by "f{z) has h (h~2) poles b jEE 3 (j=1, h
2, ••• ,h) of orders I3j (j= 1, 2, ••• ,h) respectively with ~ I3j=p and Ibj-b y I ~11 0 j=l
:::;;;;j,i':::;;;;h,j#j' ;11>0)." Here k, a,(j=1.2,···,k),
o and
h, I3j(j=1,2,"',h), 11
are given in advance; the number p will also depend on them.
8 Qm-NORMAL FAMILIES OF MEROM ORPHIC FUNCT[ONS
We have defined in Chapter 1 the notions of Co-point, Co-sequence and normal family of merom orphic functions, and in Chapter 5 the notions of CI-point, CI-sequence and quasi-normal family of meromorphic functions. Let S dn (z) (n = 1 ,2, ••• ) be a sequence of merom orphic functions in a domain D. Denote by E the set of nonCo-points of S in D and by E' the set of the points of accumulation in D of E. Then E =
0,
if S is a Co-sequence in D, and E' =
0,
if S is a CI-se-
quence in D. It is natural to go sHIl further and introduce notions corresponding respectively to the conditions E"= 0, E"'= 0
, ....
In what follows we shaU de-
fine for each integer m;;;::O, the notions Cm-point, Cm-sequence and Qm-normal family of merom orphic functions, corresponding to the condition E(m) =
0.
In
particular a Qo-normal family of meromorphic functions is a normal family of merom orphic funtions and a QI-normal family of meromorphic functions is a quasi-normal family of merom orphic functions, defined respectively in Chapter 1 and Chapter 5. Here Q is the first letter of the prefix quasi - .
8. 1. SOME NOTIONS AND THEOREMS
Definition 8. 1. Let S: fn (z) (n = 1,2,"') be a sequence of meromorphic functions in a domain D and Zo a point of D. We say that Zo is a Co-point of S, if
r : IZ-Zo I
there is a circle in
Zo is a CI-point of S, if there is a circle Iz-zo I
309
Normal F amtlies of Meromorp htc Functwns
310
al. for an integer m::;?:l. after having defined the notion of Cm_I-point of S. we say that
Zo
IZ-Zo I
is a Cm-point of S. if there is a circle
that each point of the domain 0<
Iz-zo I
belonging to D such
is a Cm_I-point of S. In this way.
the notion of Cm-point of S is defined for each integer m::;?:O.
Lemma 8. 1. If. for an integer m::;?:O.
Zo
is a Cm-point of S. then the fol-
lowing assertions hold: 1 0 Zo is a Cm+l-point of S. 2 0 There is a circle
IZ-Zo I
of this cir-
cle is a Cm-point of S. 3 0 Zo is a Cm-point of every subsequence of S. This lemma is easily proved by mathematical induction.
Definition 8. 2. m::;?: 0 being an integer. we say that a sequence S of meromorphic functions in a domain D is a Cm-sequence in D. if each point of D is a Cm-point of S.
Lemma 8. 2. If. for an integer m::;?:O. S is a Cm-sequence in D. then S is a Cm+l-sequence in D. and every subsequence of S is a Cm-sequence in D. This lemma follows immediately from Lemma 8. 1.
Definition 8. 3. Let S be a sequence of meromorphic functions in a domain D and m::;?:O an integer. A point Zo
Zo
of D is said to be a nonCm-point of S. if
is not a Cm-point of S.
Theorem 8. 1. Let S be a sequence of meromorphic functions in a domain D. If. for an integer m::;?:O. S is a Cm-sequence in D. then the set E of nonC opoints of S in D is at most enumerable.
Proof. For the case m = O. the set E is empty. For the case m = 1. the set E has no point of accumulation in D. hence by Lemma 5. 6. the set E is at most enumerable. To complete the proof. suppose that Theorem 8.1 is true for an integer m::;?:l and consider a Cm+l-sequence S of meromorphic functions in a domain D. Let a be the set of nonCm-points of S in D. Then a has no point of accumula-
Qm-normal
Fam~hes
of Meromorphw Functions
311
tion in D, and hence is at most enumerable by Lemma 5. 6. On the other hand, S is a Cm-sequence in the domain Dl =D-a. Consequently, by assumption, the set El of the nonCo-points of Sin Dl is at most enumerable. Evidently the set ECaU E 1, hence E is at most enumeralbe. Thus Theorem 8. 1 is also true for the integer m+l.
Definition 8. 4. let E be a set of points in a domain D. Denote by E' the set of the points of accumulation of E in D, by Elf the set of the points of accumulation of E' in D, in general, by E (m) the set of the points of accumulation of E(m-l) in D. If, for an integer m;?:O, the sets E(j)(j=0,1,··· ,m ;E(O)=E) are all nonempty, we say that the set E has the property W m with respect to D. Theorem 8. 1 can be precised as follows:
Theorem 8. 2. Let S be a sequence of meromorphic functions in a domain D, and m;?:O an integer. In order that S is a Cm-sequence in D, it is necessary and sufficient that the set of nonCo-points of S in D does not have the property Wm with respect to D.
Proof. The necessity of the condition can be proved as follows: For m = 0,1, the necessity is evident. Now assume that the condition is necessary for an integer m;?:l. Consider a Cm+1-sequence S of merom orphic functions in a domain D, and suppose that the set E of the nonCo-points of S in D has the property W m+ 1 with respect to D. Let Zo be a point of E(m+D and Iz-zo I
Normal
312
Fam~hes
of
Meromorph~c
Functwns
Consequently the set E of the nonCo-points of S in D has the property Wm with respect to D and the set E(m) nd is nonempty. Since r may be arbitrarily small, Zo
E E (m+D. Hence E has the property W rn+1 with respect to D. Definition 8. 5. Let 3T be a family of meromorphic functions in a domain D and m?O an integer. We say that the family 3T is Qrn-normal in D, if from every sequence of functions of the family 3T, we can extract a subsequence which is a Cm-sequence in D. 3T is said to be Qm-normal at a point Zo of D, if there is a circle
r: IZ-Zo I
belonging to D such that 3T is Qm-normal in
r.
In particular, a Qo-normal family is a normal family and a QI-normal family is a quasi-normal family, according respectively to Definition 1. 5 and Definition
5.2. Evidently according to Definition 8. 5, if 3T is Q rn-normal in D, then 3T is Qrn-normal at each point of D. Conversely we have the following theorem:
Theorem 8. 3. Let 3T be a family of meromorphic functions in a domain D and m?O an integer. If 3T is Qrn-normal at each point of D, then 3T is Qrnnormal in D.
Proof. By Theorems 1. 4 and 5. 1, we know already that Theorem 8.3 is true in the cases m = 0,1. In general, Theorem 8. 3 is proved in the same way as for Theorem 1. 4. In order to get the proof of Theorem 8. 3, it is sufficient to replace in the proof of Theorem 1. 4, the word "normal" by "Qrn-normal" the term "Co-sequence" by "Cm-sequence" and the term "Co-point" by "Crn-point".
Lemma 8. 3. If, for an integer m?O, a family 3T of meromorphic functions in a domain D is Qrn-normal in D, then 3T is Qm+l-normal in D. This lemma is an immediate consequence of Lemma 8. 2. Conversely we have the following theorem:
Theorem 8. 4. Let 3T be a family of meromorphic functions in a domain D and m?l an integer. Assume that the following conditions are satisfied:
1°3T is Qrn-normal in D. 2 0 There is a number 0 (0<0< 1) such that for each function Hz) E 3T,
Qm-normal Famzlies of Meromorphtc Functions
313
there exist two values a; (f) E C (j = 1,2) such that Ia) (f) ,a2 (f) I>b and that the function f(z) does not take the values aj (z) (j= 1,2) in D. Then the family 7
is Qo-normal in D.
Proof. By Theorem 5. 2, Theorem 8. 4 is true for the case m = 1. Now assume that Theorem 8.4 holds for an integer m;?:1. .Consider a Qm+)-normal family 7
of meromorphic functions in a domain D, satisfying the condition 2 0 in
Theorem 8. 4. Let 8 be a sequence of functions of 7. Then from 8 we can extract a subsequence 8) which is a Cm+)-sequence in D. Denote by a the set of the nonCm-points of 8) in D. By Definition 8. 1, a has no point of accumulation in D. In the domain ~ =D-a, 8) is a Cm-sequence.
A fortiori,
8) constitutes a Qm-nor-
mal family in ~. 8ince 8) satisfies the condition 2 0 in Theorem 8. 4, so by assumption, 8) is Qo-normal
in~.
From the sequence 8) we can then extract a sub-
sequence 8 2 which is a Co-sequence in
~.
Then 8 2 is a C)-sequence in D and, Ii
fortiori, constitutes a Q)-normal family in D. Again since 8 2 satisfies the condition 2 0 in Theorem 8.4, so by Theorem 5.2, 8 2 is Qo-normal in D. Finally from 8 2 we can extract a subsequence 8 3 which is a Co-sequence in D. This proves that 7
is Qo-normal in D and hence Theorem 8.4 also holds for the integer m
+ 1.
From Theorem 8. 4, we deduce the following corollary:
Corollary 8. 1. Let 8: fn (z) (n = 1, 2,,,,) be a sequence of meromorphic functions in a domain D and m;?:1 an integer. Assume that the following conditions are satisfied: 1 0 8 is a Cm-sequence in D. 2 0 There is a number b(O
Ian, ,an, I>b and that
the function fn (z) does
not take the values a n;(j=1,2) in D. Then the sequence 8 is a Co-sequence in D.
Proof. By the condition 10 in Corollary 8. 1, 8 considered as a family of functions is, Ii fortiori, Qm-normal in D. Next by the condition 2 0 in Corollary 8. 1, 8 is Qo-normal in D. On the other hand, by the condition 10 in Corollary 8. 1 and Theorem 8.1, 8 has a Co-point Zo in D. Consequently we can find a circle
Iz
-Zo I
Normal
314
Fam~hes
of Meromorphtc
Funct~ons
rem 6. 2 and conclude that S is a Co-sequence in D.
8. 2. AN EXISTENCE THEOREM In view of Lemma 8. 3, it is natural to ask the following question: Given arbitrarily an integer m?l, can we always find a family of meromorphic functions in a domain D, which is Qm-normal in D, without being Qm_I-normal in D? We are going to give an affirmative answer to this question. First of all, we point out that, for this purpose, it is sufficient to prove the following theorem:
Theorem 8. 5. Given any integer m? 1, we can find a sequence S of merom orphic functions in a domain D, such that S is a Cm-sequence in D, without being a Cm_I-sequence in D, and that the same is true for every subsequence of S. In fact, such a sequence S of meromorphic functions in a domain, considered as a family of meromorphic functions in the same domain, evidently satisfies the required conditions in the above question. To prove Theorem 8.5, consider first the case m = 1. In this case, the construction of S is known and easy. In fact, the sequence of functions
f.(z) = nz (n = 1,2,"') are hoi om orphic in C and has only one nonCo-point, namely the point z = O. Hence this is a CI-sequence in C, without being a Co-seqence in C. This is also true for each of its subsequences. In order to prove Theorem 8. 5 for the case m?2, we need the following two lemmas:
Lemma 8. 4. Let E be a set of points of C satisfying the following conditions: 1 0 E is infinite and enumerable. 2 0 E nEt is empty. Then we can construct a sequence S of holomorphic functions in C such that the set E UEt is precisely the set of nonCo-points of S in C, and that the same is true for every subsequence of S.
Qm-normal Famlhes of Meromorphic Functions
315
Proof. Since E is enumerable. it can be represented by a sequence of points an (n = 1. 2 •••• ). By condition 2 0 in Lemma 8. 4. to each point an corresponds a number O
We may suppose that bn+ 1
>
0
(n
O
=
1.2.···). hm P. 11-1+ 00
=+
00
(n=1.2.···). hmd.=O 11--1+ 00
(8.2)
(8.3)
and construct a sequence of functions S:fn (z) (n = 1.2 •.•. ) as follows:
fl(z)
= d~~l (Z-al)
f2(z)
= d~~~(z-al)(z-a2)
Consider the point al and a circle Cl: IZ-all
It follows that in Cl we have
By (8. 2) • this inequality shows that al is a nonCo-point of S. In general. consid-
Normal Famthes of Meromorphte Functions
316
er a point an and a circle cn: Iz-anl 26,-6
>
26,-6, = 6,.
It follows that, for any integer k~O, we have in cn,
(8.4)
and hence an is a nonCo-point of S. Thus we have shown that each point of E is a nonCo-point of S. Of course each point of E' (if E' is nonempty) is also a nonCo-point of S. Now consider a point zoEtEUE'. Then we can find a circle Iz-zol
c,
Let N be a positive integer such that d n
I>
P.·
Hence Zo is a Co-point of S. It is then proved that the set E U E' is precisely the set of nonCo-points of S. This is also true for any subsequence fn'(z) (j = 1,2,,,,) of S, because, from (8.4), we have in c n ,
provided that j is sufficiently large.
Lemma 8. 5. Let
p~l be an integer and a
we can find a set of points E of the closed interval [a, b ] satisfying the following
Q .. -normal Familws of Meromorphtc Functions
317
conditions:
°
1 E is infinite and enumerable.
2° EnE' is empty. 3° The set E UE' has the property W p with respect to C and (E UE' ) (p+1) is empty.
Proof. When p = 1, it is sufficient to take the sequence of points
E: x.
=
a
1 + -:;;(b-a)
(n
=
1,2",,).
(8.5)
Evidently E satisfies the conditions 1° and 2°. Moreover, we have E' = (a), (E U E')' = (E U (a»' = a, hence the condition 3° is also satisfied. Now suppose that Lemma 8.5 is proved for an integer p~1. Consider the integer p
+ 1 and a closed interval
[a, b]. Let Xn (n = 1, 2,,,,) be defined by (8.
5) and consider the sequence of closed intervals:1;: [X2pX2j_l](j= 1,2,,,,), By hypothesis, to each j corresponds a set of points E j of the closed interval Ij satisfying the conditions 1°,2°,3° for the integer p. Consider the set of points E = U E j j=l
of the closed interval [a, b]. It is easy to see that E satisfies the conditions 1° and 2°. On the other hand, it is also easy to see that
E'=(UE/)U(a) j=l
which implies that
E
U
(UH)
E'
U
(a),
j=l
where Hj=E j UE;'. Since HjCI p we see that
(E U E')' = (UH j j=l
)'
U (a) = (UH/) U (a). j=l
318
Normal Familtes of Meromorphic Functions
Then we have successively 00
(E UE')"= (UH;) j-I
(E
U E'
00
)(p)
=
(UH/'»
U (a), U
(a).
j=l
Since H j (,+1) is empty(j=1,2,"'), we see that (E
U
E' )(,+1) = (a).
Consequently the set E UE' has the property W.+ I with respect to C and (E U
E' )(,+2) is empty. Therefore Lemma 8.5 is also true for the integer p+l. Now let us return to the proof of Theorem 8.5 for the case m~2. Given an integer m~2, consider the integer p =m-l~l. By Lemma 8. 5, we can find a set of points E of C, satisfying the conditions 1 0 ,2 0 ,3 0 in Lemma 8.5. From the first two conditions, by Lemma 8. 4, we can construct a sequence S of holomorphic functions in C such that the set E UE' is precisely the set of nonCo-points of Sin C and that the same is true for every subsequence of S. Since the set E UE' satisfies the condition 30 in Lemma 8. 5, then by Theorem 8. 2, S is a C.+I-sequence in C, without being a C.-sequence in C, and the same is true for every subsequence of S. Since p+ l=m, p=m-l, S satisfies the conditions required by Theorem 8. 5 with D=C.
8.3. NECESSARY AND SUFFICIENT CONDITIONS FOR Qm-NORMALITY Definition 8. 6. Let S: fn (z) (n = 1,2,,,,) be a sequence of meromorphic functions in a domain D and Zo a point of D. We say that Zo is a Ill-point of S, if for each circle
f: IZ-Zo I~r belonging to D, we have ltm maxJ(z,f.) =+00.
11-4+= zEI
Zo is called a Il 2-point of S, if for each circle
r : IZ-Zo I
Qm-normal Famtltes of Meromorphic Functions
319
~1-point z' in the domain O
called a ~m-point of S, if for each circle
r : IZ-Zo I
~m-l
point in the domain 0< IZ-Zo I
~1-point
of S, then Zo is a
of S'. By mathematical induction, we see that in general, m?:1 being
an integer, if Zo is a
~m-point
of S, then Zo is a
~m-point
of S' .
Theorem 8. 6. Let ST be a family of meromorphic functions in a domain D. In order that ST be Q a-normal (or normal) in D, it is necessary and sufficient that every sequence S: fn (z)(n= 1,2, ••• ) of functions of ST has no
~ I-point
in D.
Proof. By Theorem 1. 6, the condition is evidently necessary. To prove that the condition is also sufficient, assume that it is satisfied and consider a sequence S dn (z)(n = 1, 2, ••• ) of functions of ST. Then the sequence of functions
a
(z,fn)(n=1,2,···) must be locally uniformly bounded in D. In fact, otherwise there is a point Zo E D such that we can not find a circle
r : IZ-Zo I
D and a number M >0 such that for n?:l, we have a(z
in
r.
,f.)
~ M
Take two sequences of numbers r.>0(k=1,2,···) and M.>0(k=1,2,···)
such that
lim r.
1-,)+=
and that the circles
=
0, ltm M. .-')+=
=+
00
r.: IZ-Zo I~r. (k = 1 ,2, ••• ) belong to D.
teger nl?:l such that
Next there is an integer n2>n] such that
T hen there is an in-
Normal Families of Meromorpliic Funct,ons
320
Continuing in this way, we get successively a sequence of integers nt (k = 1 , 2, ···;nt+l>nt) such that for k~l, we have
Consider a circle
f: IZ-Zo I~r belonging to D. Let
ko~l be an integer such that
rt
hence lim maxJ(z ,f.)
i-i+oo .Ef
=+
00.
t
Zo is then a Ill-point of the sequence of functions fn,. (z)(k = 1,2, ••• ), incompatible with the hypothesis. So we have shown that the sequence of functions J(z,fn)(n=1,2,···) is lo-
cally uniformly bounded in D. Consequently by Theorem 1.6, the sequence S: fn(z) (n=1,2, ... ) constitutes a Qo-normal family in D and hence we can extract from S a subsequence S' which is a Co-sequence in D.
Definition 8. 7.
m~l being an integer, let $T be a family of meromor-
phic functions in a domain D and
v~O
an integer. We say that $T is Qm-normal
of order vat most in D, if from every sequence of functions of the family $T, we can extract a subsequence which is a Cm-sequence in D and has at most v nonCm-I-points in D. In particular when v=O, $T is Qm_I-normal in D.
Theorem 8.7. Let D and
v~O
$T be a family of meromorphic functions in a domain
an integer. In order that $T be QI-normaI(or quasi-normal) of order
vat most in D, it is necessary and sufficient that every sequence S: fn (z) (n = 1 , 2, ••• ) of functions of $T has at most v Ill-points in D.
Proof. In the case v=O, Theorem 8.7 reduces to Theorem 8.6. Consider
Q .. -normal Families of Meromorphic Functions
321
the case v~1. Assume that the family Y is Q)-normal of order vat most in D. Let S:fn (z)(n = 1,2, ••• ) be a sequence of functions of Y. Suppose, on the con~)-points
trary, that S has v+ 1
Zj(j= 1,2, ••• , v+ 1) in D. By hypothesis, from S
we can extract a subsequence S' which is a C)-sequence in D and has at most v nonCo-points in D. Denote by E the set of nonCo-points of S' in D. Then in the domain D' =D-E, S' is a Co-sequence and hence constitutes a Qo-normal family. Among the points zj(j=1,2,···,v+1), there is at least one, for instance z)E D'. By Theorem 8.6, S' has no ~)-point
~)-point
in D'. But on the other hand, z) is a
of S'. So we get a contradiction.
Next assume that the condition of Theorem 8.7 is satisfied. Let S be a sequence of functions of Y. If every subsequence of S has no
~)-point
in D, then
by Theorem 8. 6, S constitutes a Qo-normal family in D, hence we can extract from S a subsequence So which is a Co-sequence in D. Otherwise S has a subsequence S' which has a
~)-point
z) in D. If every subsequence of S' has no
~)
point in the domain D - (z), then we can extract from S' a subsequence S) which is a Co-sequence in D- (z). Otherwise S' has a subsequence S" which has a
~)-point
Z2 in D- (z). Continuing in this way, after at most v steps, we get a
subsequence S(Y)ofS, such that S(y) has v~)-points zj(j=1,2,···,v) in D. In the domain D - (z) ,Z20 ••• ,Zy), every subsequence of S(y) has no ~)-point, hence we can extract from S(y) a subsequence Sy which is a Co-sequence in D - (z), Z20 ••• ,
z.). Theorem S.S.
Let Y be a family of merom orphic functions in a domain
D. In order that Y be Q)-normal (or quasi-normal) in D, it is necessary and sufficient that every sequence S:fn(z) (n=1,2,···) of functions of Y has no ~2point in D.
Proof. Assume that the family Y is Q I-normal in D. Let S be a sequence of functions of Y. Then from S we can extract a subsequence S' which is a C)sequence in D. Suppose, on the contrary, that S has a
~2-point
Zo in D. Since Zo
is a C)-point of S' , we can find a circle Iz-zo I
~)-point
~)-point
in d. But by Definition 8.6, S has a
of S' , we arrive at a contradiction.
~)-point
z, in d. Since z'
Normal
322
Fam~ltes
of
Meromorph~c
Functions
Now assume that the condition of Theorem 8.8 is satisfied. Consider a circle
T:
Iz-zo 1~r belonging to D. By Theorem 8.3, it is sufficient to show that
the family $T is Q1-normal in the circle T : 1Z-Zo 1 +1) is a subsequence of S(k).
2) For each k~I, S(k) has a Ill-point z. in T. 3) The sequence of points Zk (k = 1,2, ... ) are distinct. Let S' be the diagonal sequence of the sequences S (k) (k = 1,2, ... ). Then each point Zk is a Ill-point of S' , because beginning from the k-th term of S' , all the following terms of S' belong to S(k). Let Zo be a limiting point of the sequence of points Zk (k = 1, 2, ... ). Then Zo E rand Zo is a 1l2-point of S' , incompatible with the hypothesis that the condition of Theorem 8. 8 is satisfied. In general we have the following two theorems:
Theorem 8. 9. m~I being an integer, let $T be a family of meromorphic functions in a domain D and v~O an integer. In order that $T be Qm-normal of order vat most in D, it is necessary and sufficient that every sequence S: fn (z) (n= 1, 2, ... ) of functions of $T has at msot v Ilm-points in D.
Theorem 8. 10. m~I being an integer, let $T be a family of merom orphic functions in a domain D. In order that $T be Qm-normal in D, it is necessary and sufficient that every sequence S dn (z) (n = 1,2, ... ) of functions of $T has no Ilm+1-point in D.
Proof. From Theorems 8.7 and 8.8, we know that Theorems 8.9 and 8. 10 are true for m= 1. Assume that Theorems 8.9 and 8. 10 hold for an integer m ~1.
Let us first prove the following theorem: Let $T be a family of merom orphic functions in a domain D and v~O an inte-
323
Qm-normal Families of Meromorphic Functions
ger. In order that .57 be Qm+l-normal of order vat most in D, it is necessary and sufficient that every sequence of functions of .57 has at most v ~m+l-points in D. For the case v=O, this theorem reduces to Theorem 8.10. Consider the case v;?: 1. Assume that .57 is Qm+l-normal of order v at most in D. Let S be a sequence of functions of.57. Suppose, on the contrary, that S has v+ 1 ~m+l-points Zj (j = 1,2, ... , v+ 1) in D. From S we can extract a subsequence S' which is a Cm+l-sequence in D and has at most v nonCm-points in D. Denote by E the set of nonCm-points of S' in D. Then in the domain D' =D-E, S' is a Cm-sequence and hence constitutes a Qm-normal family. By Theorem 8.10, S' has no
~m+l
point in D'. But among the points zj(j=1,2,"',v+1) there is at least one, for instance ZI ED'. Since ZI is also a
~m+l-point
of S' , we get a contradiction.
Conversely assume that the condition of the above theorem is satisfied. Let S be a sequence of functions of.57. If every subsequence of S has no ~m+l-point in D, then by Theorem 8. 10, S constitutes a Qm-normal family in D, hence we can extract from S a subsequence So which is a Cm-sequence in D. Otherwise S has a subsequence S' which has a
~m+l-point
ZI in D. If every subsequence of S'
has no ~m+l-point in D - (ZI)' then we can extract from S' a subsequence SI which is a Cm-sequence in D- (ZI)' As in the proof of Theorem 8.7, continuing in this way, after at most v steps, we get a subsequence S(v) of S, such that S (v) has
v~m+l-points
zj(j=1,2,"',v) in D. In the domain D-(ZpZ2,''',zv), every
subsequence of S (v) has no ~m+l-point, hence we can extract from S (v) a subsequence Svwhich is a Cm-sequence in D-(ZI,Z2,"·,Z.). Next let us prove the following theorem: Let .57 be a family of merom orphic functions in a domain D. In order that .57 be Qm+l-normal in D, it is necessary and sufficient that every sequence of functions of .57 has no
~m+2-point
in D.
Assume that .57 is Qm+l-normal in D. Let S be a sequence of functions of .57. Then from S we can extract a subsequence S' which is a Cm+l-sequence in D. Suppose, on the contrary, that S has a
~m+2-point
Zo in D. Since Zo is a C m+ l-
point of S' , we can find a circle Iz-zo I
so a
in d. But by Definition 8. 6, S has a
~m+l-point
~m+l-point
Z, in d. Since z, is al-
of S' , we get a contradiction.
Conversely assume that every sequence of functions of .57 has no
~m+2-point
Normal
324
in D. Consider a circle
Fam~he8
of
Meromorph~c Funct~on8
r: IZ-Zo I~r belonging to D.
By Theorem 8. 3, it is suf-
ficient to show that the family $T is Qm+l-normal in the circle r : Iz-zo I
~m+l-point
of S'. Let Zo be a limiting point of the sequence of points
Z. (k = 1,2, ••• ). Then Zo E rand Zo is a
~m+2-point
of S' , incompatible with the
hypothesis. In what follows, we deduce from Theorems 8. 7 - 8. 10, some generalizations of Marty's theorem. For this we need to give the following definitions:
Definition 8. 8. Consider a domain D. Let points of D. We say that
a(1)
a(1)
:z;(l~j~n) be a system of
satisfies the condition (r 1) with respect to D, if ei-
ther n=l or n>l and the n points z;(l~j~n) are distinct. Let a(l):zj(j;;:'1) be a sequence of points of D, we say that
a(1)
satisfies the condition (r'l) with respect
to D, if the points z; (j;;:'1) are distinct and the limit lim z;= A exists, with A=A H+=
(aO»ED. Let a(2)
a(2)
:zi;(l~i~n ,j;;:'1) be a double sequence of points of D. We say that
satisfies the condition (r 2) with respect to D, if for each i(l~i~n), the se-
quence
aW : a(2) :
aP) :Zi;(j;;:'1)
A.=A(ai 1»
satisfies the condition (r;) with respect to D and the system
(l~i~n) satisfies the condition (r 1) with respect to D. Let
Zij 0;;:'1 , j;;:' 1) be a double sequence of points of D. We say that am satisfies
the condition (r~) with respect to D, if for each i 0;;:'1), the sequence a/I) :zlj(j ;;:'1) satisfies the condition (r;) with respect to D, and the sequence a(l): A, = A (ail)
0;;:'1) satisfies the condition (r;) with respect to D.
Qm-normal Famihes of Meromorphic Functions In general, let
a(m)
325
:zi,i,".J,. (1~jl~n ,h~l, ••• ,jm~1) be a m-ple (m~3) se-
quence of points of D. We say that
a(m)
satisfies the condition (r m) with respect
to D, if for each system jlOh, ••• ,jm-I (1~jl~n ,j2~1, ••• ,jm-I~1), the sequence a~l;... im_' :zitj..··J,._,im(jm~1) satisfies the condition (r;) with respect to D, and the
(m -1)-ple sequence Ai,j .... jm_' = A(aj~l;... J,,_) (1~jl~n ,h~l, ••• ,jm-I~I) satisfies the condition (r m-I) with respect to D. Let
a(m):
Zi,i,-.J" (jl~1 ,j2~1, .•• ,jm~1) be
a m-ple (m~3) sequence of points of D. We say that
a(m)
satisfies the condition
(r:n) with respect to D, if for each system jlOj2,···,jm-l(jl~l,j2~1,···,jm-l~ 1), the sequence ai~l;... im_': Zitj.... J"_,J,, (jm ~ 1) satisfies the condition (r;) with re-
spect to D, and the (m-1)-ple sequence Aj,j.... J,,_,=A(aj~1;... J,,_)(jl~I,j2~1,···, jm-I~1) satisfies the condition (r:n-I) with respect to D.
Definition 8. 9. Let
sr be a family of meromorphic functions
in a domain
D and Zj(j= 1,2, ••• ,n) a system of points of D. We say that the family
sr satis-
fies the condition (M) with respect to the system zj(j=I,2,···,n), if there exist circles
Tn IZ-Zj I~ri(j= 1,2, ••• ,n)
belonging to D and a number A>O such that
for each function f(z) E sr, we have
min (maxJ(z ,f) l:S;;;;j~
~
A.
zEIJ
sr be a family of meromorphic functions in a domain D and v~O an integer. In order that sr be QI-normal (or quasi-normal) of order v at most in D, it is necessary and sufficient that sr satisfies the condition (M) Theorem 8. 11. Let
with respect to every system of points Zj(1~j~v+ 1) of D satisfying the condit on (r I) with respect to D.
Proof. For the case v= 0, Theorem 8. 11 reduces to Theorem 1. 6. Consider the case v~1. First assume that
sr is
QI-normal of order vat most in D.
Let Zj (1 ~j~ v+ 1) be a system of distinct points of D. Suppose, on the contrary, that
sr does
not satisfy the conditon (M) with respect to the system Zj(1
~j~v+l). Take a number p>O such that the circles Iz-zjl~p (1~j~v+1)
belong to D. Then to each integer n~l, corresponds a function fn (z) E sr such that
Normal Families of Meromorphic Functions
326
min {maxJ(z,f.)}>n, l~j~v+l
zEr;a>
where nn) denotes the circle Iz-zj I::::;;1:.. Hence n
maxJ(z,f.»n O::::;;j::::;;v+1). zEl~·)
Consider the point
ZI
and a circle
T: IZ - Z I I::::;;r
belonging to D. Let no~ 1 be an
integer such that for n~no we have p/n
maxJ(z,f.)
~maxJ(z,f.)
>n
zEl~·)
zEl
and hence
ltm maxJ(z,f.) _-)+00
This shows that ••. ). Similarly
Zj
ZI
zEI
=+
00.
is a ill-point of the sequence of functions S: fn (z) (n = 1,2,
(j = 2 , ••• ,v+ 1) are also ill-points of S. Consequently S has v+ 1
ill-points in D. But by Theorem 8. 12 and the hypothesis, S has at most v illpoints in D. So we get a contradicition. N ext assume that the condition of Theorem 8. 11 is satisfied. Let S: fn (z) (n
= 1,2, ••• ) be a sequence of functions
of.'i7. By Theorem 8. 7, it is sufficient to
show that S has at most v ill-points in D. Suppose, on the contrary, that S has v + 1 ill-points zjO::::;;j::::;;v+ 1) in D. Take arbitrarily a system of circles
Tj: Iz-zj I
::::;;rjO::::;;j::::;;v+1) belonging to D and a number A>O, then by Definition 8.6, we have
Consequently we can find an integer
N~1
such that
Q .. -normal Famthes of Meromorphic Functions
327
This contradicts the hypothesis that the condition of Theorem 8. 11 is satisfied.
Theorem 8. 12. Let gr be a family of meromorphic functions in a domain D. In order that gr be Q I-normal (or quasi-normal) in D, it is necessary and sufficient that for every sequence ZjG;:;:'1) of points of D satisfying the condition (r;) with respect to D, we can find an integer p;:;:'l such that gr satisfies the
condition (M) with respect to the system
Zj(1:,,:;:}~p).
Proof. First assume that gr is QI-normal in D. Let z/j;:;:'1) be a sequence of points of D satisfying the condition (r;) with respect to D. Suppose, on the contrary, that we can not find an integer p;:;:' 1 having the required property. Take a circle Ill: Iz-zl I ~rl~l belonging to D, then there is a function fl (z) E gr such that
maxJ(z,fl)
>
1.
zEIll
Next take two circles I 2j : Iz-zjl~r2~1/2 (j=1,2) belonging to D, then there is a function f 2(z) E gr such that mtn (max J(z ,f 2»
1~j:S;;;2
zEIzj
>
2.
In general, take p circles Ip;: Iz-zjl~rp~l/p (j=1,2,· .. ,p) belonging to D, then there is a function fp (z) E gr such that mtn (maxJ(z,f,» l~i~p
zEl,j
>
p.
Obviously for any point Zj and any circle I: IZ-Zj I~r belonging to D, we have hm maxJ(z,f,) 1-1+ 00 zEI
=+
00.
So by Definition 8. 6, the points Zj (j;:;:' 1) are all ~I-points of the sequence S: fp (z) (p = 1, 2, ... ) of functions of gr. Since the sequence of points Zj(j;:;:'1) satisfies the condition (r;) with respect to D, Zj converges to a point I.E D. A is then
328
Normal Families of Meromorphic Functions
a Ilz-point of S. But by the hypothesis and Theorem 8.8, S has no Ilz-point in D. Thus we get a contradiction. N ow conversely assume that the condition of Theorem 8. 12 is satisfied. Let S: fn (z) (n = 1,2, ••• ) be a sequence of functions of $T. By Theorem 8. 8, it is sufficient to show that S has no Ilz-point in D. In fact, suppose, on the contrary, that S has a Ilz-point I;, in D. Then there is a sequence of distinct Ill-points Zj{j~ 1) of S, such that lim Zj = 1;,. The sequence of points Zj(j~1) satisfies then the H+oo
condition (r;) with respect to D. It is easy to see that for this sequence of points Zj(j~1),
we can not find an integer
p~1
having the required property in Theo-
rem 8. 12, incompatible with the hypothesis.
Theorem 8. 13. Let $T be a family of meromorphic functions in a domain D and v~O an integer. In order that $T be Qz-normal of order vat most in D, it is necessary and sufficient that for every double sequence z'j(l~i~v+ 1 ,j~l) of points of D satisfying the condition (r z) with respect to D, we can find an integer p~1 such that $T satisfies the condition (M) with respect to the system Z'j (l~i~v+l,l~j~p).
Proof. For the case v= 0, Theorem 8. 13 reduces to Theorem 8. 12. Consider the case v~l. First assume that $T is Qz-normal of order vat most in D. Let a(2):zij(l~i~v+l,j~1) be a double sequence of points of D satisfying the condition (r z) with respect to D. Suppose, on the contrary, that we can not find an integer p~1 having the required property. p~1 being an integer, take a number r p(O
rij: Iz-zijl~rp(l~i~v+l,
l~j~p)
belong to D, then there is a function fp (z) E $T such that m~n l~i:S;;;:v+l
l~j~,
(maxJ(z,f,))>p. zEl. IJ
It is easy to see that each point Zlj of
S:fp(z) (p=I,2,···). Since
a(Z)
a(2)
is a Ill-point of the sequence of functions
satisfies the condition (r z) with respect to D,
for each i(l~i~v+1), the limit lim Zlj = 1;,1 exists and the points 1;,,(l~i~v+1) j~+oo
are distinct and belong to D. The points
I;,,(l~i~v+ 1)
are then Ilz-points of S,
incompatible with the hypothesis, by Theorem 8. 9. Now assume that the condition of Theorem 8.13 is satisfied and let us show
329
Q .. -normal Families of Meromorphic Functions
that gr is Q 2-normal of order v at most in D. Let S: fn (z) (n = 1 ,2, ••• ) be a sequence of functions of gr. By Theorem 8. 9, it is sufficient to show that S has at most v 1l2-points in D. In fact, suppose, on the contrary, that S has v+ 1 112points 1;;,(l~i~v+1) in D. Then for each i (l~i~v+1), we can find a sequence Zij(j~1)
of distinct Ill-points of S, such that
In this way we get a double sequence
0(2)
:z'j(l~i~v+ 1 ,j~1) satisfying the con-
dition (r 2) with respect to D. Since the points z'j(l~i~v+1,j~1) are a11lllpoints of S, it is easy to see that we can not find an integer p~l such that gr satisfies the condition (M) with respect to the system zij(l~i~v+ 1, l~j~p), incompatible with the hypothesis.
Theorem 8. 14. Let gr be a family of meromorphic functions in a domain D. In order that gr be Q2-normal in D, it is necessary and sufficient that for every double sequence Zij (j ~ 1 ,j ~ 1) of points of D satisfying the condition (r~) with respect to D, we can find an integer
p~l
such that gr satisfies the condi-
tion (M) with respect to the system zij(l~i~p, l~j~p).
Proof. First assume that gr is Q2-normal in D. Let
o(2):zij(j~l,j~1) be
a double sequence of points of D satisfying the condition (r~) with respect to D. Suppose, on the contrary, that we can not find an integer p ~ 1 having the required property. Then as in the proof of Theorem 8. 13, we can find a sequence of functions S d p (z) (p = 1 ,2, ••• ) of gr such that each point Z'j of of S. Since
0(2)
0(2)
is a Ill-point
satisfies the condition (r~) with respect to D, by Definition 8. 8,
for each i (j~1), the limit lim zij=l;;, exists, l;;iED and lim I;;,=Z exists with ZE H+=
H+=
D. The points 1;;, (j~1) are then 1l2-points of Sand Z is a Ils-point of S. This is incompatible with the hypothesis, by Theorem 8.10. Now conversely assume that the condition of Theorem 8.14 is satisfied and let us show that gr is Q2-normal in D. Let S dn(z) (n = 1,2, ••• ) be a sequence of functions of gr. By Theorem 8.10, it is sufficient to show that S has no Ils-point in D. In fact, suppose, on the contrary, that S has ails-point Z in D. Then S has a sequence of distinct 1l2-points
1;;,(j~1)
in D such that
330
Normal Famtltes of Meromorphic Functwns ltm
i--1+=
So
=
Z.
Next for each i (i;?::l) , there is a sequence of distinct ~l-points Zii(j;?::l) in D of S, such that ltm
Zij
=
Si.
j-)+""
In this way we get a double sequence (r~)
a(2): zii
Ci;?::l, j;?:: 1) satisfying the condition
with respect to D. Since the points ZiiCi;?::1 ,j;?::l) are all ~l-points of S, it is
easy to see that for the double sequence
a(2)
we can not find an integer p;?::1 hav-
ing the required property in Theorem 8. 14. So we get a contradiction with the hypothesis. By the same method we can prove the following two general theorems:
Theorem 8. 15. m;?::2 being an integer, let $T be a family of meromorphic functions in a domain D and v;?::O an integer. In order that $T be Qrn-normal of order v at most in D, it is necessary and sufficient that for every m-ple sequence Zi\i,"im (1~jl~V+ 1 ,j2;?::1, ••• ,jrn;?::1) of points of D satisfying the condition (r rn) with respect to D, we can find an integer p;?::1 such that $T satisfies the
condition (M) with respect to the system zi\i,"im (1~jl~V+ 1, 1~j2~P"'" 1~jrn ~p).
Theorem 8.16. m;?::2 being an integer, let $T be a family of meromorphic functions in a domain D. In order that $T be Qm-normal in D, it is necessary and sufficient that for every m-ple sequence Zj\i,"'i. (jl;?:: 1, j2;?:: 1, "', jrn;?:: 1) of points of D satisfying the condition (r~) with respect to D, we can find an integer p;?:: 1 such that $T satisfies the condition (M) with respect to the system Zi\i,"'im (1~jl~P' 1~j2~P'''' ,1~jrn~P)'
8.4. CONDITIONS AND CRITERIONS OF Qm-NORMALITY
Definition 8. 10. m;?:: 0 being an integer, a condition K is said to be a condition of Qrn-normality, if it has the following properties: 10 If a meromorphic function Hz) in a domain D satisfies the condition K in
Qm-normal Families of Meromorphic Functions
331
D, then fez) satisfies the condition K in every subdomain of D. 2 0 If each function of a family
J}T
of meromorphic functions in a domain D
satisfies the condition K in D, then the family
Definition 8. 11.
J}T
is Qm-normal in D.
m~l and v~O being two integers, a condition K is said
to be a condition of Qm-normality of order vat most, if K has the property lOin Definition 8. 10 and has the following property: 2 0 If each function of a family
J}T
of merom orphic functions in a domain D
satisfies the condition K in D, then the family
J}T
is Qm-normal of order vat most
in D. Consider a domain D and a finite number of circles
r j Iz-zj I
p) such that their closures f;: Iz-zjl~rj(j=1,2, .. ·,p) belong to D. We say that the circles
r j (j=1,2,"',p) are distinct, if f j (j=1,2, .. ·,p) are mutually dis-
joint.
Definition 8. 12.
m~O being an integer, let K be a condition of Qm-nor-
mality and E>O a number. Let f(z) be a meromorphic function in a domain D. A
r:
circle
Iz -Zo I
following properties: 1 of: Iz-zo I ~r belongs to D. 2 0 r<E. 3 0 The function f(z) does not satisfy the condition K in
Theorem 8. 17.
m~O being an integer, let K be a condition of Qm-nor-
malityand E>O a number. Let
J}T
be a family of meromorphic functions in a do-
main D. If each function fez) of the family family
J}T
r.
J}T
has no (K ,e)-circle in D, then the
is Qm-normal in D.
Proof. Consider a point Zo of D and let r : Iz-zo I
r.
J}T
satisfies the condition K in
Since Zo is arbitrary,
Theorem 8. 18. Let
J}T
J}T
r.
Hence the family
J}T
is Qm-normal
is Qm-normal in D by Theroem 8. 3.
be a family of meromorphic functions in a domain
332
Normal Families of Meromorphtc Functtons
D. m?:O being an integer, let K be a condition of Qm-normality and e>O a number. Let v?:O be an integer. If each function f(z) of (K , e)-circles in D, then
.'j}T
.'j}T
has at most v distinct
is Qm+l-normal of order vat most in D.
In the case v= 0, by Theorem 8. 17 the family
.'j}T
is Qm-normal in D, hence
the conclusion of Thoerm 8.18 holds. In the case v>O, Theorem 8. 18 is proved in the same way as for Theorem 5.3 in the case v>O. It is sufficient to replace in the proof of Theorem 5.3, the word "normal" by "Qm-normal", the term "Co-sequence" by "Cm-sequence", the term "CI-sequence" by "Cm+l-sequence" and the term "nonCo-point" by "nonCmpoint" , in order to get the proof of Theoerm 8. 18. Denoting by K' the condition of Theorem 8.27, namely "f(z) has at most v distinct (K ,e)-circles in D", then by Theorem 8.27, K' is a condition of Qm+lnormality of order vat most. The condition K' depends on m,K,e and v. We express this dependence in writing K' = £J(m,K ,t,v).
(8.6)
Now let Ko be a condition of Qo-normality. Let em>O (m = 0,1,2, ••• ) be a sequence of numbers and vm?:O (m=0,1,2,···) a sequence of integers. Then
is a condition of Q I-normality of order Vo at most;
is a condition of Q2-normality of order VI at most; in general, for m?:l,
is a condition of Qm-normality of order Vm-I at most. We may say that the sequence of conditions Km (m = 1 ,2, ••• ) is generated by the condition K 0 and the two sequences €m(m=0,1,2,···) and vm(m=0,1,2,···). N ext we give some generalizations of Theorems 3. 9,5. 4,3. 17 and Corollary 5. 7. For this we need the following definition:
Qm-normal
Fam~hes
of
Meromorph~c
333
Functions
Definition 8.13. Given a sequence of integers II=l,Im;?2 (m=2,3, .. ·) and a sequence of numbers Tlm
>0
tion Hz) in a domain D, a circle
(m = 1 , 2, ... ), consider a merom orphic func-
r : IZ-Zo I
a (a) I-circle of the function Hz) in D, if
value a E C. We say that
r
is
following properties:
AI)!: Iz-zol~r belongs to D. B I) r
r
r.
is a (a)2-circle of the function f(z) in D, if
r
has the property
AI) and the following properties: B2) r
r
r.
is a (a)m-circle of the function
r has the property AI) and the following properties:
Bm) r
r.
Lemma 8. 6. Let m;?l be an integer, 6(0<6<1) a number and N;?l an integer. Let 57 be a family of meromorphic functions in a domain D. Assume that for each function f(z)E57, there are three values aj=aj(f)EC (j=1,2,3) such that (8.7)
and that for each j
(I~j~3)
Hz) has at most N distinct (aj)m-circles in D. Then
the family 57 is Qm-normal of order 3N at most in D.
Proof. In the case m = 1, Lemma 8. 6 follows from Corollary 5. 3. Now assume that Lemma 8.6 is true for an integer m;?1. Then given a number 6(0< 6<1) and an integer N;?l, we deduce from Lemma 8.6 a corresponding condition K m (6,N) which is a condition of Qm-normality of order 3N at most. In order to complete the proof of Lemma 8. 6, it is sufficient to show that the condition K m + 1 (o,N) is a condition of Qm+l-normality of order 3N at most. To see this, let f (z) be a meromorphic function in a domain D and assume that there are three values ajEC (j=1,2,3) such that (8.7) holds and that for each j(I~j~3) Hz)
Normal Families of Meromorphic Functions
334
has at most N distinct (aj)m+l-circles in D. Consider the condition K' = Km (0, I m + 1 ) , where Im+ 1 is given in Definition 8. 13. We know already that K' is a condition of Qm-normality of order 3Im+ 1 at most. Suppose that the function f(z) has a (K' ,llm+l)-circle
r
in D, where 1lm+l is given in Definition 8. 13. Then by Defi-
nitions 8.12 and 8.13, we see that among the three values aj(j=1,2,3), there is at least one value aj such that f(z) has I m+ 1
+1 distinct
(aj)m-circles in
r,
hence
r
is a (aj)m+l-circle of f(z) in D, by Definition 8. 13. It follows that the function f (z) has at most 3N distinct (K' , 1lm+l )-circles in D. Consequently the condition K m+ 1 (0 ,N) is a condition of Qm+l-normality of order 3N at most, by Theorem 8.
18.
Theorem 8. 19. Let
m~O be an integer and 0(0<0<1) a number. Let
!iT be a family of merom orphic functions in a domain D. Assume that for each function f(z)E!iT, there are three values aj=a/f)EC (j=1,2,3) such that (8.
n
holds and that for each j (I~j~3) f(z) has no (aj)m+l-circle in D. Then the
family !iT is Qm-normal in D. This theorem is a generalization of Theorem 3. 9 which corresponds to the case m=O.
Proof. Assume such that
T:
m~1. Consider a point zoED and a circle
r:
Iz-zol
Iz-zol~r belongs to D and r where llm+l is given in Defini-
tion 8.13. Let f(z)E!iT. Then for each j(I~j~3) f(z) has at most I m+ 1 distinct (a)m-circles in otherwise,
r
r,
where I m+ 1 is given in Definition 8. 13 and aj =aj (f), because
would be a (aj)m+l-circle of f(z) in D, incompatible with the hy-
pothesis. Consequently by Lemma 8. 6, the family !iT is Qm-normal of order 3Im+ 1 at most in
r.
Since the point Zo is arbitrary, hence by Theorem 8.3, the
family !iT is Qm-normal in D. In what follows we shall try to obtain a generalization of Theorem 5.4. For this we need first to give some definitions and prove some lemmas.
Definition 8. 14.
m~l being an integer, let S:fn(z) (n=1,2,···) be a
Cm-sequence of meromorphic functions in a domain D. Assume that S has a nonCm_1-point Zo in D. Then two cases are possible: 1 0 We can extract from the sequence S a subsequence S' of which Zo is a
Qm-normal Famihes of Meromorphic Functions
335
Cm_I-point. In this case, we say that S is reducible with respect to the point zoo 2 0 Zo is a nonCm_l-point of every subsequence of S. In this case, we say that S is irreducible with respect to the point zoo The sequence S is said to be an irreducible Cm-sequence in D, if S has nonC m_ l points in D with respect to each of which S is irreducible.
Lemma 8.7. m?I being an integer, let S: fn(z) (n=1,2,···) be a Cm-sequence of meromorphic functions in a domain D. Assume that S has nonC m- l points in D. Then we can extract from S a subsequence S' which is either a Cm-I-sequence in D or an irreducible Cm-sequence in D. This lemma is proved in the same way as for Lemma 5. 5. In order to get the proof of Lemma 8. 7, it is sufficient to replace in the proof of Lemma 5. 5, the term "CI-sequence" by "Cm-sequence", the term "Co-point" by "Cm-I-point" and the term "nonCo-point" by "nonCm_l-point".
Definition 8. 15. Let S be a sequence of meromorphic functions in a domain D. S is called a completely irreducible CI-sequence in D, if S is an irreducible CI-sequence in D. S is called a completely irreducible C 2 -sequence in D, if S satisfies the following conditions: 1) S is an irreducible C 2 -sequence in D.
2) E being the set of nonCI-points of S in D, S is a completely irreducible CI-sequence in the domain D-E. In general, m? 2 being an integer, S is called a completely irreducible Cm-sequence in D, if S satisfies the following conditions:
l' ) S is an irreducible Cm-sequence in D. 2' ) E being the set of nonCm_l-points of S in D, S is a completely irreducible Cm-I-SeqUence in the domain D-E.
Lemma 8. 8. m?I being an integer, let S be an irreducible Cm-sequence of meromorphic functions in a domain D. Then we can find a subsequence S' of S, such that S' is a completely irreducible Cm-sequence in D.
Proof. By Definition 8. 15, Lemma 8. 8 is trivial in the case m = 1. Consider the case m = 2. Let S be an irreducible C 2-sequence of merom orphic functions
336
Normal Families of Meromorphic Functions
in a domain D. E being the set of nonCI-points of Sin D, S is a CI-sequence in the domain G = D - E. S must have nonCo-points in G, because otherwise, S would be a Co-sequence in G and each point of E would be a CI-point of S. So by Lemma 8. 7, we can extract from S a subsequence S' which is either a Co-sequence in G or an irreducible CI-sequence in G. S' can not be a Co-sequence in G, because S is irreducible with respect to each point of E. Hence S' is an irreducible CI-sequence in G. Moreover evidently S' is also an irreducible C 2-sequence in D. Thus Lemma 8.8 is proved in the case m=2. Now suppose that Lemma 8.8 is true for an integer m?::2 and let us show that it is also true for m
+ 1.
In fact, let S be an irreducible Cm+l-sequence of
meromorphic functions in a domain D. E being the set of the nonCm-points of S in D, S is a Cm-sequence in the domain G = D - E. In G, S has necessarily nonCm-I-points. By Lemma 8.7, we can extract for S a subsequence SI which is either a Cm_l-sequence in G or an irreducible Cm-sequence in G. The first alternative can not occur, because each point of E is a nonCm-point of SI. So SI is an irreducible Cm-sequence in G. Hence, by assumption, we can extract from SI a subsequence S' which is a completely irreducible Cm-sequence in G. Since S' is a subsequence of S, S' is an irreducible Cm+l-sequence in D and its set of nonC mpoints is E. Consequently S' is a completely irreducible Cm+l-sequence in D.
Lemma 8.9. m?::l being an integer, let S: fn(z) (n=1,2,···) be a completely irreducible Cm-sequence of meromorphic functions in a domain D and let E be the set of nonCm_l-points of Sin D. Then we can find a value woEC having the following properties: 1 0 For any point zoEE, any circle I: Iz-zol
11--1+ 00
we can find a circle I that, when n?::N, I
I
Iw. , W I =
0, W =1= w 0'
(8.8)
: I z - Zo I
I
2° For any point Zo E E, any circle I : IZ-Zo I
I
is a (w)m-circle of the function fn(z) in I.
337
Q .. -normal Families of Meromorphic Functions
Proof. By Definition 8. 15 and Lemma 5.7, Lemma 8. 9 is true in the case m = 1. Now assume that Lemma 8. 9 is true for an integer m~1 and let us show that it is also true for m
+1.
In fact, let S: fn (z) (n = 1,2, .. ·) be a completely ir-
reducible Cm+l-sequence of meromorphic functions in a domain D and let E be the set of nonCm-points of SinD. By Definition 8. 15, S is a completely irreducible Cm-sequence in the domain G=D-E. Let EI be the set of nonCm-1-points of Sin G. By assumption, we can find a value Wo E e having the following property: For any point Zo E Ep any circle T : IZ-Zo I
lim
_--1+
Iv.,v I = 0,
V =F woo
00
Consider a circle c': Iz-/;,ol
T j: I z - Zj I
T; : Iz-zj I
when n~Ni'
T; is a (vn)m-circle of the function fn(z) in T j' Set
Then when
n~N,
the function fn(z) has Im+1 distinct (vn)m-circles
T; (j=1,2,
... ,lm+l) in c' , hence c' is a (vn)m+l-circle of the function fn(z) in the circle c. Next consider again a point /;,oEE, a circle c: Iz-/;,o I
n~N,
c' is a (v)m+l-circle of the function fn(z) in the circle c.
This completes the proof that Lemma 8. 9 is also true for the integer m N ow we are in a position to prove the following theorem:
+ 1.
Normal Families of Meromorphic Functions
338
Theorem 8. 20. Let
m~l be an integer, b(O
1,2,3) three nonnegative integers such that PI ~P2~P3. Let.'iT be a family of meromorphic functions in a domain D. Assume that for each function fez) E.'iT, there are three values aj(f) E
C (j= 1,2,3)
such that
and that for each j (1~j~3), fCz) has at most pj distinct (aj(f) )m-circles in D. Then the family .'iT is Qm-normal of order P2 at most in D. This theorem is a generalization of Theorem 5. 4 which corresponds to the case m=1.
Proof. Let
N~l be an integer such that P3~N. Then by hypothesis and
Lemma 8. 6, the family .'iT is Qm-normal of order 3N at most in D. Consider a sequence 8 of functions of .'iT. We have to show that from 8 we can extract a subsequence 8' which is a Cm-sequence in D and has at most P2 nonCm-l-points in D. For this we first extract from 8 a subsequence 8 1 which is a Cm-sequence in D and has at most 3N nonCm_l-points in D. Assume that 8 1 has nonCm_l-points in D. Then by Lemma 8. 7, we can extract from 8 1 a subsequence 8 2 which is either a Cm-I-sequence in D or an irreducible Cm-sequence in D. Assume that 8 2is an irreducible Cm-sequence in D. Then by Lemma 8.8, we can extract from 8 2 a subsequence 8 3 which is a completely irreducible Cm-sequence in D. Let 8 3 be the sequence of functions fn (z) (n = 1, 2, ••• ) and set a)') = aj(f,) (n = 1,2,···)() = 1,2,3).
By Lemma 3. 2, we can find an increasing sequence of positive integers nq (q = 1, 2, •.. ) such that for each j(1~j~3), there is a value Aj E
C such
with
IA
j'
A j'
I~6
(1 ~ j , j' ~
3, j =I=- j' ).
that
Q.. -normal Families of Meromorphic Functions
339
Set g.(z)=fn.(z) (q=1,2,"') and
It is easy to see that the sequence of functions S' :g.(z) (q=1,2,"') is also a
completely irreducible Cm-sequence in D. Moreover we have
and for each jO~j~3) ,g.(z) has at most pj distinct (13}'»m-circles in D. Applying Lemma 8. 9 to the sequence of functions S' , we conclude that there is a value Wo E
C having
the property 1 0 in that lemma, with respect to S' .
One at least of the two values Aj(j= 1,2) is different from wo, for example A2-=P woo Assume that S' has ~ nonCm_1-points z.(k=1,2,,,·,~) in D. Let I.: Iz-z. l
be ~ circles belonging to D and disjoint with each other.
Then for each kO~k~~) we can find a circle I~: Iz-z.l
at most P2 nonCm-1-points in D. In what follows, we give some generalizations of Theorem 3. 17 and Corollary 5.7.
Lemma 8. 10. Let N~1 ,v~1
m~1 be an integer, M ,b(O
two integers. Let $T be a family of meromorphic functions in a do-
main D. Assume that for each function f(z) E $T, there are two values a (f) and b (f) such that Ia (f)
I<
M, 15 <
Ib (f) I <
M
(8.9)
and that f(z) has at most N distinct (a (f) )m-circles in D and f(v) (z) has at most N distinct (b (f»m-circles in D. Then the family $T is Qm-normal of order 2N at most in D.
Normal
340
Fam~hes
of Meromorphic Functwns
Proof. By Corollary 5.7 and Definition 8.13, Lemma 8.10 is true in the case m=1. Now assume that Lemma 8. 10 is true for an integer m?::1. Then given two numbers M, b (O
r
in D, where llm+l is given in Definition 8. 13.
Then by Definition 8. 12 and Definition 8. 13, either fez) has Im+l (f»rn-circles in
r,
hence
r
1 distinct (b (f) )rn-circles in
+ 1 distinct
is a (a(f»rn+l-circle off(z) in D, or f(v)(z) has I m+ 1
r,
hence
r
(a
+
is a (b (f) )m+l-circle of f(v) (z) in D. Con-
sequently the function f (z) has at most 2N distinct (K' , llm+l )-circles in D. Hence the condition Km+l (M ,b, N ,v) is a condition of Qm+l-normality of order 2N at most, by Theorem 8.18.
Theorem 8. 21. Let m?::O be an integer, M ,000
be a family of merom orphic functions in a domain D.
Assume that for each function f(z) E 7, there are two values a (f) and b (f) such that (8.9) holds and that f(z) has no (a(f»m+l-circle in D and tv)(z) has no (b (f) )m+l-circle in D. Then the family 7
is Qm-normal in D.
This theorem is a generalization of Theorem 3.17 which corresponds to the case m=O.
Proof. Assume m?::1. Consider a point zoED and a circle r: Iz-zol
T:
Iz-zol~r belongs to D and r
tion 8. 13. Let f(z) E 7. Then f(z) has at most Im+l distinct (a (f»m-circles in
r , where Irn+l is given in Definition 8. 13, because otherwise, r would be a (a (f»m+l-circle of f(z) in D. Similarly f(v) (z) has at most Im+l distinct (b (f»m-circles in
r. Hence r. Since
most in
by Lemma 8.10, the family 7
is Qm-normal of order 2Irn+l at
the point Zo is arbitrary, hence by Theorem 8.3, the family 7
Qm-normal Families of Meromorphic Functwns
341
is Qm-normal in D.
Theorem 8. 22. Let ~O ,q~O
m~l be an integer, M ,b(O
two integers and v~l an integer. Let $r be a family of meromorphic
functions in a domain D. Assume that for each function Hz) E $r, there are two values a(f) and b(f) such that (8.9) holds and that Hz) has at most p distinct (a(f»m-circ1es in D and tv)(z) has at most q distinct (b(f»m-circles in D. Then the family $r is Qm-normal of order p+q at most in D. This theorem is a generalization of Corollary 5. 7 which corresponds to the case m=1.
Proof. Assume
m~2.
Consider the condition K' =Km-1(M,b,Im,v) de-
duced from Lemma 8. 10, which is a condition of Qm-l-normality of order 21m at most, where 1m is given in Definition 8. 13. Let Hz) be a function of the family $r, so that there are two values a(f) and b (f) such that (8.9) holds and that f
(z) has at most p distinct (a(f»m-circles in D and f(v)(z) has at most q distinct (b(f»m-circ1es in D. Suppose that Hz) has a (K' ,llm)-circle T in D, where llm is given in Definition 8.13. Then either Hz) has I m+1 distinct (a(f»m-l-circ1es in
r,
hence
r
is a (a (f) )m-circ1e of Hz) in D, or f(v) (z) has 1m + 1 distinct (b
(f) )m-l-circ1es in T , hence
r is a (b (f»m-circle of f(v) (z) in D. Consequently f
(z) has at most p +q distinct (K' 'llm )-circles in D and, by Theorem 8. 18, the family $r is Qm-normal of order p+q at most in D.
8.5. OTHER CRITERIONS OF Qm-NORMALITY In what follows we give some criterions of Qm-normality of a different kind. Consider a domain D and a nonempty set E of points of D. Let m~l be an integer and assume that E does not have the property W mwith respect to D, according to Definition 8. 4. Let us investigate the question: Under what condition the set D-E is a domain? First if m=l, then it is easy to see that D-E is a domain. Assume m~2 and E'
#0.
We are going to show that in order that D-E is a
domain, it is necessary and sufficient that E' CEo In fact, assume that D-E is a domain. Let zoEE' and suppose, on the contrary, that zoEtE. Then since E' CD, we have zoED-E. Hence there is a circle
r : Iz -
Zo
I< r
Hence E' CEo
such that
r CD -
E, incompatible with the condition Zo E E' .
Normal Families of Meromorphic Funct,ons
342
Conversely assume that E' CEo Consider first the case m=2, so that by hypothesis, E"=0. Then the set G=D-E' is a domain. Consequently, if E= E' , then D-E is a domain. Assume E#E'. Define E1=E-E' , then D -
E
=
G -
E 1•
Consider a point zoEG. Then we can find a circle
r:
Iz-zol
such that the domain 0< I z - Zo I
+ 1-
In fact, assume that E does not have the property W m+1 with respect to D, and distinguish two cases: DE does not have the property W m with respect to D, then by assumption, D-E is a domain. 2) E has the property Wm with respect to D. Then E(m)::f:0, E(m+1) = 0. Then eE' ) (m-I) =E (m)::f: 0, eE' ) (m) = 0. Consequently the set E' does not have the property W m with respect to D. On the other hand, from the condition E' C E, we deduce eE')' CE'. Hence, by assumption, the set G=D-E' is a domain. Then by the same procedure used in the case m=2, we conclude that DE is a domain.
Theorem 8. 23. Let ST be a family of meromorphic functions in a domain D. Let E be a nonempty set of points of D and m~l an integer. Assume that the following conditions are satisfied:
1 0 E does not have the property W m with respect to D and the set D- E is a domain. 2 0 The family ST is Qo-normal in the domain D-E. Then the family ST is Qm-normal in D.
Proof. Let S be a sequence of functions of the family ST. Then by the condition 2 0 in Theorem 8. 23, we can extract from S a subsequence S' which is a Co-sequence in D-E. Denote by a the set of nonCo-points of S' in D. Then aC E and hence by the condition lOin Theorem 8.23, a does not have the property W m with respect to D. Consequently by Theorem 8. 2, S' is a Cm-sequence in D.
Q .. -normal Families of Meromorphic Functions
343
More generally we have the following theroem:
Theorem 8. 24. Let $T be a family of meromorphic functions in a domain D. Let E be a nonempty set of points of D and m;?l.k;?O two integers. Assume that the following conditions are satisfied: 1 0 E does not have the property W mwith respect to D and D - E is a domain. 2 0 The family $T is Qt-normal in the domain D-E. Then the family $T is QmH-normal in D. For the proof of this theorem. we need the following lemmas:
Lemma 8. 11. Let D be a domain and m;?l.k;?O two integers. Let E be a nonempty set of points of D. such that E does not have the property W m with respect to D and that D-E is a domain. Let a be a set of points of the domain D -E. such that a does not have the property W t with respect to D-E. Then a does not have the property W mH with respect to D.
Pr oof. Denote by a(j) (D) (j = 1, 2 ... ·) the successive derived sets (not necessarily nonempty) of a with respect to D. and by a(j) (D - E) (j = 1. 2 .... ) the successive derived sets of a with respect to D - E. Suppose. on the contrary. that a has the property W mH with respect to D. so that the sets a(j) (D) (j = 0 .1 • .. ·.m+k;a(O)(D)=a) are all nonempty. It is easy to see successively that a' (D) a"(D)
n (D -
n (D
-
E) C a' (D E) C a"(D -
E) • E) •
By hypothesis. a(t) (D - E) is empty. hence a(t) (D)
n (D -
E) is empty and a(t)
(D)CE. Since a(k+O(D)(j=O.I ... ·.m) are all nonempty. a(t)(D) has the property Wm with respect to D. On the other hand. since a(t)(D)CE and E does not have the property W m with respect to D. a(t) (D) can not have the property W m with respect to D. Thus we get a contradiction.
Lemma 8.12. Let D be a domain. E j (j=1.2) two sets of points of D and
Normal Families of Meromorphic Functions
344
mj;;::::O (j=1,2) two integers. Assume that Ejdoes not have the property W m , with respect to D, for j= 1,2. Then the set E =EI UE2 does not have the property W p with respect to D, where p=max(ml ,m2)'
Proof. Denote by E' ,E",··· and E; ,E;,··· respectively the successive derived sets of E and E j in D. It is easy to see that the relation E =EI UE2 always implies the relation
E' = E;
U E~,
whether E; is empty or not. Then we have successively
N ow suppose, on the contrary, that the set E has the property W p with respect to D. Then, by definition, the sets E(;)(j=O,l,···,p) are all nonempty. In particular E (p) is nonempty. Hence one at least of the two sets E?) (j = 1,2), say E(p), is nonempty. This implies that the sets E(J)(j=O,l,···,p) are all nonempty. Since ml~p, EI has the property W m, with respect to D, incompatible with the hypothesis. Now let us prove Theorem 8.24. Let S be a sequence of functions of the condition
20
sr.
By
in Theorem 8. 24, we can extract from S a subsequence S' which
is a C.-sequence in the domain D-E. Denote by in D-E, then by Theorem 8.2,
0
0
the set of nonCo-points of S'
does not have the property W. with respect to
D-E. Hence by Lemma 8.11, o does not have the property W mH with respect to D. Next by Lemma 8.12, the set EUodoes not have the property W mH with respect to D. Denote by E Uo and hence
01
01
the set of the nonCo-points of S' in D, evidently
01 C
does not have the property W mH with respect to D. Finally by
Theorem 8. 2, S' is a CmH-sequence in D. In Theorems 8. 23 and 8. 24, the set E is fixed and independent of the function f(z) E sr. In what follows, we extend Theorem 8.24 to the case that the set E is variable with the function f(z) E sr.
Theorem 8. 25. Let
sr be a family of meromorphic functions
D. Let {a} be a collection of nonempty sets
0
in a domain
of D and m;;::::l ,k;;::::O two integers.
Q .. -normal Fam,lws of Meromorphic Functions
345
Let K be a condition of Qk-normality. Assume that the following conditions are satisfied: 1 0 For each set aE {a}. the set D-a is a domain. 2 0 For every sequence of sets an (n = 1. 2 •.•. ) of {a}. we can find a subsequence an, (p = 1 .2 •.•• ) and a nonempty set E of points of D. having the following properties: a) E does not have the property Wm with respect to D and the set D-E is a domain. b) For each point zoED-E. there exist a circle r: Iz-zol
Proof. Let S:fn(Z) (n=I.2.···) be a sequence of functions of the family 57. Then by the condition 2 0 in Theorem 8. 25. we can find a subsequence an, = a(fn,) (p = 1. 2 •••• ) of the sequence of sets an =a(fn) (n = 1. 2 •••• ) and a nonempty
set E of points of D having the properties a) and b). Consider the subsequence Sf :fn,(z)(p=I.2.···) of S and a point zoED-E. Then by the property b).
there exist a circle r : IZ-Zo I
dition 3 0 • the function fn, (z) satisfies the condition K in the circle r. This implies that the family of functions fn, (z) (P?:Po) and hence the family of functions Sf :fn, (z) (p= 1. 2 •••• ) is Qk-normal in r. Since the point zoE D-E is arbitrary,
we conclude that the family Sf is Qk-normal in the domain D-E. Consequently. by Theorem 8. 24 the family Sf is QmH-normal in D and hence we can extract from Sf a subsequence Sit which is a CmH-sequence in D.
9 APIPLICATIONS OF THE THEORY OF Qm-NORMAL FAMIILIES OF MEROM ORPHIC FUNCTIIONS
9. 1. UNIFORM CONVERGENCE OF SEQUENCES OF MEROMORPHlC FUNCTIONS Theorem 9. 1. Let S: fn (z)(n = 1,2, ... ) be a sequence of holomorphic functions in a domain D satisfying the following conditions: 1 0 For an integer m?: 1, S belongs to a Qm-normal family of holomorphic functions in D. 2 0 There exists a set E of points 011' D, such that Ifor the same integer m, E has the property W m with respect to D (see Definition 8. 4), and that at each point of E, S converges to a finite limit. Then S converges locally uniformly to a holomorphic function in D. This theorem tis a generalization of Theorem 6. 1 which corresponds to the case m = 1. For the proof of this theorem, we need the foHowing lemma:
Lemma 9. 1. Under the conditions of Theorem 9. 1, we can extract from the sequence S a subsequence S' which converges locally unilformly to a holomorphic function tin D.
Pr oof. For the case m = 1, this lemma has already been proved in the proof of Theorem 6. 1. To complete the proof of Lemma 9. 1, suppose that it is true Ifor an integer m?:l, and consider the case corresponding to the integer m
+
1. In this case, we can extract from the sequence S a subsequence S' which is a
347
348
N ormal
Fam~hes
of
Meromorplt~c
Functwns
Cm+l-sequence in D. Let <J be the set of nonCm-points of S' in D. Then S' is a Cm-sequence in the domain DI =D-<J and, Ii fortiori, S' considered as a family of functions is Qm-normal in D I. By considering a point zoEE(m+I), we see as in the proof of Theorem 6. 1, that there is a point ZI E E (m) that the set EI =E
nDI has
nD I.
Then it is easy to see
the property Wm with respect to D I. Moreover S'
converges to a finite limit at each point of E I' Hence, by assumption, we can extract from S' a subsequence S" which is locally uniformly convergent in D I . Then, as in the proof of Theorem 6. 1, we see that S" is locally uniformly convergent in D. The proof of Lemma 9. 1 is now complete. Now let us prove Theorem 9. 1. First by Lemma 9. 1, we can extract from the sequence S a subsequence S' which converges locally uniformly to a holomorphic function F (z) in D. Then, as in the proof of Theorem 6. 1, we can show that the sequence S converges locally uniformly to the function F (z) in D.
Definition 9.1. Let E be a set of points in a domain D, and
m~O,v>O
two integers. We say that E has the property W m" with respect ot D, if E has the property Wm with respect to D and the set E(m) consists of at least v points.
Corollary 9. 1. Let
m~2 and v~O be two integers. Let S: f n (z)(n=1,2,
••• ) be a sequence of holomorphic functions in a domain D satisfying the following conditions: lOS belongs to a family .5JT of holomorphic functions in D, which is Qm-normal of order vat most in D. 2 0 There exists a set E of points of D, such that E has the property Wm_I,,+1 with respect to D, and that at each point of E, S converges to a finite limit. Then S converges locally uniformly to a holomorphic function in D.
Proof, By the condition 10 , we can extract from the sequence S, a subsequence S' which is a Cm-sequence in D and has at most v nonCm_l-points in D, Denoting by e the set of nonCm_l-points of S' in D. Then in the domain DI =D-e,
S' is a Cm_I-SeqUence, S' considered as a family is then Qm_I-normal in D I. By the condition 2 0 , the set E (m-I) consists of at least v+ 1 points. Hence there is a point
nE (m-I).
r : IZ-Zo I
Consider a circle
has the property W m_1 with respect to Dl' Moreover at each point of E I , S' con-
Apphcatwns of the Theory of Qm-normal Famthes
349
verges to a finite limit. Consequently by Theorem 9. 1, Sf converges locally uniformly to a holomorphic function in Dl" Next we see easily that Sf converges locally uniformly to a holomorphic function F (z) in D. Finally we see that the sequence S converges locally uniformly to the function F (z) in D, by the usual method.
Theorem 9. 2. Let S: fn (z) (n = 1,2, ••• ) be a sequence of meromorphic functions in a domain D satisfying the following conditions: 1 0 For an integer m;;?:l, S belongs to a Qm-normal family
sr of meromorphic
functions in D. 2 0 S has no exceptional point in D. 3 0 There exists a set E of points of D, such that for the same integer m, E has the property W mwith respect to D, and that at each point z E E , S converges (with respect to the spherical distance) to a limit A(Z) such that for every value a EC, the set {z IzEE ,A(z)=a} does not have the property Wm with respect to D. Then S is a Co-sequence in D and the limit function of S (with respect to the spherical distance) is a non-constant meromorphic function in D. This theorem is a generalization of Theorem 6. 3 which corresponds to the case m = 1. For the proof of this theorem, we need the following lemma:
Lemma 9. 2. If for an integer m;;?:O, a sequence of meromorphic functions fn (z)(n = 1, 2, ••• ) in a domain D is a Cm-sequence in D, then for any complex number c, the sequence fn(z)+c(n=1,2,···) is a Cm-sequence in D. Moreover if fn(z)~O
in D(n=1,2,···), the sequence 1/fn (z)(n=1,2,···) is also a Cm-se-
quence in D.
Proof. By Lemmas 5. 2 and 5. 3, Lemma 9. 2 is true in the cases m = 0 and m = 1. By mathematical induction, we can prove Lemma 9. 2 in general, by basing upon Definitions 8. 2 and 8. 1Now let us return to the proof of Theorem 9.2. Denote as usual by E(J)(j= 1,2, ••• ) the derived sets of E in the domain D. Then by the condition 3 0 , the sets E (j) (j = 0,1,2, ••• , m) are nonempty. Consider a point Zo E E (m) and let us show that Zo is a Co-point of the sequence S :fn (z) (n= 1,2, ••• ). By the condition 2 0 , there exist a circle T : IZ-Zo I
Normal Families of Meromorphic Functwns
350
r. Define for n~N , f.(Z) if a = F .(z)
=
{
00
1
(9.1)
f.(z)-a if a lS flnite.
Then by Lemma 9.2, the sequence Fn(z)(n~N) belongs to a Qm-normal family of holomorphic functions in r. (This is because that, by the condition 1 0 , the sequence {fn(z) (n~N)) constitutes a Qm-normal family in r. Then by Lemma 9. 2, this is also true for the sequence {Fn(z)(n~N)}.) On the other hand, the set E!=Enr has the property Wm with respect to r, and by the condition 30, the set
e)
=
{zlz E EIOA(Z) =a}
(9.2)
does not have the property Wm with respect to r. Since
the set E)-e! has the property Wm with respect to r , by Lemma 8.12. Moreover at each point of E)-elO the sequence Fn(z)(n~N) converges to a finite limit. It follows that we can apply Theorem 9. 1 to the sequence of holomorphic functions Fn(z)(n~N)
in r and the set E)-e)Cr, and conclude that the sequence Fn(z)(n
~N) converges locally uniformly to a holomorphic function in r. (z)(n~N)
A fortiori,
Fn
is a Co-sequence in r. Then by Lemma 9.2, the sequence S :fn (z)(n
= 1,2",,) is a Co-sequence in r. Hence Zo is a Co-point of S. Now we are going to show that each point of D is a Co-point of S. Consider a point z)ED(z!7~zo). Join zoand z) by a polygonal line Z(t)(O~t~l) lying in D with Z(O)=zo and ZO)=ZI' Since Zo is a Co-point of S, there is a number ,(0< ,<1) such that each point of the polygonal line Z(t )(O~t~,) is a Co-point of S. Let,. be the least upper bound of the set of such numbers ,. We have 0<,. ~ 1. Then each point of the polygonal line Z(t)(O~t<,.) is a Co-point of S. To
each t (O~t<,.) corresponds a circle r.: Iz-Z(t) I
Appltcatwns of the Theory of Qm-normal Famtlws d
351
U I,.
O~t
It is easy to see that d is a subdomain of D and S is a Co-sequence in d. Let Hz)
be the limit function (with respect to the spherical distance) of Sin d. On the set EI,we have Hz)=A(z). By the condition 3°, A(Z) is nonconstant in E I . Consequently Hz) is a nonconstant merom orphic function in d. Consider the point z. =Z(-r.). Again by the condition 2°, there exist a circle
r • : Iz-z.
I
that for each n~N • , the function fn (z) does not take the value a. in
r .' Evi-
dently we can find a line segment L:Z(t)(tl~t~tHO
r •.
L considered as a set of points has the property W mwith respect to
r • , and
the set
{zlz E L,f(z) = a.} consists of at most a finite number of points. Consequently, as for the point zo, we conclude that the point z. is a Co-point of S. It follows that we can not have ,.<1. Therefore ,.=1 and the point ZI=Z(l) is a Co-point ofS. Thus we have proved that S is a Co-sequence in D. Since, by hypothesis, the limit function (with respect to the spherical distance) of S is nonconstant on the set E, it must be a nonconstant meromorphic function in D. In what follows, we shall give an application of Theorem 9. 1. For this we need the following definition:
Definition 9. 2. Let Hz) be a meromorphic function in a domain D. Let
r: Iz-zol
r.
r is a generalized (a)2-circle of the function Hz) in D, if r has the
property PI) and the following property: Q2) The function Hz) has at least 12 distinct generalized (a)l-circles in where 12 is given in Definition 8. 13. In general, that
r
m~2
is a generalized (a)m-circ1e of the function f(z) in D, if
PI) and the following property:
r,
being an integer, we say
r
has the property
Normal
352
Fam~lies
of
Meromorph~c
Functions
Qm) The function f(z) has at least 1m distinct generalized (a)m_l-circles in
r,
where 1m is given in Definition 8. 13.
Lemma 9.3. Let w=cz be a linear function which maps a domain D onto a domain G, where c#O is a constant. Let f( w) be a meromorphic function in G and f(cz) the corresponding meromorphic function in D. Let
r : I Z-Zo I
be a
circle, a E C a value and m ~ 1 an integer. If the circle function f(cz) in D, then the image circle
r is a (a )m-circle of the r.: IW-Wo I
by the mapping w=cz is a generalized (a)m-circle of the function f(w) in G.
Proof. Consider first the case m = 1. Since, by Definition 8. 12, the circle
f:
IZ-Zo I ~r belongs to D, evidently the circle
since the function f(cz) takes the value a in a in f. Hence by Definition 9.2,
r
f:
IW-Wo I~p belongs to G. Next
r , the
function f( w) takes the value
is a generalized (a)l-circle of the function f
(w) in G. So Lemma 9. 3 is true in the case m = 1. Now assume that Lemma 9.3 is true for an integer show that it also holds for the integer m
+ 1.
m~l,
we are going to
In fact, suppose that the circle
r :
IZ-Zo I
f:
IZ-Zo I ~r belongs to D, and hence the circle
f:
IW-Wo I~p belongs to
G. Consider a circle A: Iz-z. I
r. Since the function w =cz maps the circle r : IZ-Zo I
f: IW-Wo
I
(a)m-circles in f. Consequently
r
is a generalized (a)m+l-circie of f(w) in G.
Theorem 9. 3. Let f(z) be a holomorphic function in a domain
Q defined
by Q: Izl>ro, a<argz<S (ro>O,O~a<S~2lt), and let m~1 be an integer.
Let wand
WI
be two bounded domains defined respectively by
w:ro< and
Izi
App licatiolls of the Theory of Q.. -lIormal Families
353
Ctl. :rl < Iz 1< RIt al < argz < PI (ro
Let E be a set of points of Ctl.
Ctl. ,
such that E has the property W m with respect to
Let k=Rl/rl and Ej(j~l integer) be the set of the values of the function kjz
on E. Define
(f
=
UEj
(Eo
j-O
=
E).
Assume that the following conditions are satisfied: 1 0 There are a number p>ro and two finite values a and b (a=Fb) such that in the domain 0 1 defined by
&'J:lzi >p, a<argz
as Izl---1+oo,
:Izl >ro,d <argz
zEO' , Hz) tends to L.
This Theorem is a generalization of a theorem of Montel. (see[36])
Proof. Consider an integer
n~l and the linear function knz. Evidently in
the domain ()), the set of the values of this function is the domain
Ctl,:k'ro< Izl p. Then for n~N , belongs to the domain 0 n~N,
1,
Consequently by Lemma 9.3 and the condition
10 ,
Uln
for
the function fn(z)=f(knz) has no (a)m+l-circle and no (b)m+l-circle in ()).
Of course fn(z) has no (oo)m+l-circle in ()). Hence by Theorem 8.19, the family of functions fn (z)(n~N) is Qm-normal
lfl ()).
The family fn (z)(n = 1,2,,,,) is also
Normal Famtlies of Meromorphic Functions
354
Qm-normal in w. On the other hand, by the condition 2 0 , at each point of the set E, the sequence fn(z)(n=1,2,· .. ) converges to the limit L. Hence by Theorem 9. 1, the sequence fn (z) (n = 1,2, ••• ) converges locally uniformly to a holomorphic function F(z) in w. Evidently F(z)=L in w. Consider a domain
Wi :rl
Wi
defined by
~ Iz I ~ RIO a' < argz <
/1 ' (a <
a' <
/1' < /1).
Then the sequence fn (z) (n = 1,2, ••• ) converges uniformly in
Wi
to the constant
L. Evidently this implies that as Izl-1+ oo , zEQI , Hz) tends to L.
9. 2.
DISTRIBUTION OF THE VALUES OF MEROM ORPHIC
FUNCTIONS IN THE NEIGHBORHOOD OF AN ESSENTIAL SINGULARITY.
Lemma 9. 4. Let S: fn (z) (n = 1,2, ••• ) be a sequence of meromorphic functions in a domain D, Zo a point of D and m ~ 1 an integer. Let Dl CD be a subdomain of D, such that Zo E D 1 • Assume that Zo is a J.lm-point of S. Then S considered as a sequence of meromorphic functions in D 1 , Zo is also a J.lm-point of
S. Proof. By Definition 8. 6, evidently Lemma 9. 4 is true in the case m = 1. Then by mathematical induction, we see that Lemma 9. 4 holds in general, in basing upon Definition 8. 6. For the sake of convenience we introduce several notations which will be used in what follows. Consider a linear function cz(c::;t:O) and a circle Iz-zol
F (zo, r ; c) the circle Iz-cZo I ~ I c I r which is the set of the values taken by the function cz in the circle IZ-Zo I~r. Consider a merom orphic function Hz) in the finite part of a domain Iz I >ro (ro>O). Let D be a domain in a bounded domain ro
Apphcatwns of the Theory of Qm-normal Famtltes
355
(9.3)
and consider the sequence of functions Sd,(z) = f(c,z)(n = 1,2,"')
(9.4)
meromorphic in the domain D. We are going to find necessary and sufficient conditions in order that the sequence S considered as a family of functions, is not Qo-normal in D, or not QI-normal of order vat most in D, or not QI-normal in D, etc. , and then deduce consequences on the distribution of the values of the function Hz) in the neighborhood of the point
00.
1) Assume that the family S is not Qo-normal in D. Then by Theorem 8.6,
we can find a subsequence:
s' d"
(z) =
f
1,2,,,,)
(c.,z) (p
(9.5)
of the sequence S, such that S' has a Ill-point Zo E D. According to Definition 8. 6, for each circle IZ-Zo l:S;;;e belonging to D, we have hm P--1+OO
max <J(z,f. )
-lz-Z:ol~l'
'
=+ 00.
(9.6)
Thus we have shown that in order that the family S is not Qo-normal in D, it is necessary that we can find a subsequence S' defined by (9.5) of the sequence S and a point Zo E D such that (9. 6) holds for any circle Iz-zo l:S;;;e belonging to D. Evidently this condition is also sufficient in order that the family S is not Q o-normal in D. From this result we deduce the following theorem:
Theorem 9.4. In order that the family S is not Qo-normal in D, it is necessary and sufficient that we can find an increasing sequence of positive integers np (p = 1, 2 , ... ) and a poin t Zo ED, such that for any circle IZ-Zo l:S;;;e belonging to D, we have hm
max
1--1+00 zE"f(Zo'l',C.,)
Izl<J(z,f)
=+00.
(9.7)
356
Normal Famlhes of Meromorpllle Funetwns
Proof. First of all, in the domain D we have a(z,f.) = le"la(e.,z,f). Consider a point Z of the circle IZ-Zo I ~E. Then cn,z is a point of F (zo, E;C n) and
1 a(z,f.) ~ -Ie. z la(e. z,f), , To' , hence
(9.8)
Conversely consider a point z of F(zo,e;cn). Then z=cn,z' , where z' is a point of the circle IZ-Zo I ~E, and
Iz la(z ,f) = le.,z' la(e.,z' ,f) ~ = Ra(z'
Ric., la(e.,z' f)
,f.),
hence (9.9)
(9.8) shows that (9.6) implise (9.7). Conversely (9.9) shows that (9.7) implies (9.6). Consequently Theorem 9.4 follows from the above result. Now assume that the condition (9.7) is satisfied. Then (9.6) holds for any circle IZ-Zo I ~E belonging to D. This implies that the sequence S' defined by (9. 5) considered as a family is not Qo-normal at the point zoo Moreover every subsequence S" of S' considered as a family is also not Q o-normal at the point zoo Consider a circle Iz-zo I <e belonging to D and a number 6(0<6<1). Then by Theorem 3.9, we can find an integer
Po~l
such that for each
P~Po,
in the
circle IZ-Zo I <e, the function fn, (z) takes every value wEe except at most the
Apphcations of the Theory of Q .. -normal Families
357
values w which belong to two spherical circles Iw,wjl~o(j=1,2). Consequently for each p~po' in the circle r(ZO,EiCn,)' the function fCz) takes every value w
EC
except at most the values w which belong to two spherical circles Iw,wjl~
0(j=1,2). The circles r (ZO, e; cn) (p~po) are therefore filling circles of the function f (z).
2)Assume that for an integer v~O, the family S defined by (9.2) is not QInormal of order vat most in the domain D. Then by Theorem 8. 7, we can find a subsequence Sf defined by (9. 5) of S, such that Sf has v+ 1 J..ll-points z;Cj= 1,2, .•. , v+ 1) in D. Hence for anyone Zo of the points Zj (j = 1,2, ••• , v+ 1) and any circle Iz-zol~e belonging to D, (9.6) holds. Thus in order that the family S is not Q I-normal of order vat most in D, it is necessary that we can find a subsequence Sf defined by (9. 5) of Sand v+ 1 points Zj (j= 1, 2, ••. ,v+ 1) in D ,such that for anyone Zo of the points Zj (j= 1,2, ••. , v+ 1) and any circle I Z-Zo I ~e belonging to D, (9.6) holds. Evidently by Theorem 8. 7, this condition is also sufficient in order that the family S is not Q 1normal of order v at most in D. By means of (9.8) and (9.9), we deduce from this result the following theorem:
Theorem 9. 5. In order that the family S is not QI-normal of order vat most in the domain D, it is necessary and sufficient that we can find an increasing sequence of positive integers n p (p=1,2,···) and v+l points zJ(j=1,2,···,v+ 1) in D, such that for anyone Zo of the points Zj (j = 1,2, ••• , v+ 1) and any circle
IZ-Zo I ~e belonging to D, we have (9. 7). Now assume that the condition of Theorem 9.5 is satisfied. Then by Theorem 8. 7, the sequence Sf defined by (9. 5), considered as a family is not Q I-normal of order vat most in D, and this is also true for every subsequence S" of Sf . Denoting by
~(D,cn,)
the domain which is the set of the values taken by the
function cn,z in the domain D, then by Theorem 5. 4, we see that given arbitrarilya number 0(0<0<1), we can find an integer Po~l such that when p~po, in the domain
~
(D, cn,)' the equation fez)
=
W
Normal Families of Meromorphic Functions
358
has at least v+ 1 distinct roots for every value wEe, except the values w which belong to two spherical circles Iw,wjl~0(j=1,2). The domains
~
(D ,c n)
(p~Po)
are filling domains of the function Hz).
3) Assume that the family S defined by (9.4) is not Ql-normal in the domain D. Then from the proof of Theorem 8. 12, we see that we can find a subsequence S' defined by (9.5) of the sequence S and a sequence of points Zj(j~1) of the domain D, satisfying the condition (r'l) with respect to D, such that the points
Zj(j~1)
z,(j~1)
are all ill-points of the sequence S'. By Definition 8.8, the points
are distinct and the limit lim Zj = I. exists with I. ED. Hence by Definij--1+=
tion 8. 6, I, is a il2-point of S' . Thus in order that the family S is not Ql-normal in D, it is necessary that we can find a subsequence S' defined by (9.5) of S and a sequence of points Zj(j ~ 1)
of D, satisfying the condition (r'l) with respect to D, such that the points Zj
(j~1)
are all ill-points of S'. Evidently by Theorem 8. 12 this condition is also
sufficient in order that the family S is not Q I-normal in D. By means of (9.8) and (9.9), we deduce from this result the following theorem:
Theorem 9. 6. In order that the family S is not Ql-normal in the domain D, it is necessary and sufficient that we can find an increasing sequence of positive integers n. (p = 1, 2,,,') and a sequence of points
z;Cj~1)
of D, satisfying the
condition (r;) with respect to D, such that for anyone Zo of the points zj(j~l) and any circle Iz-zo I~e belonging to D, we have (9.7). Now assume that the condition of Theorem 9.6 is satisfied. Then the lim Zj j-)+=
= I,
exists and I,E D is a il2-point of the sequence S' defined by (9. 5). Consider a
circle I z-I,I <e belonging to the domain D. By Lemma 9. 4, S' considered as a sequence of meromorphic functions in the circle IZ-I. I <e, I. is also a il2-point of S'. Consequently by Theorem 8.8, S' considered as a family, is not Ql-normal in the circle Iz-I,I <e. This is also true for any subsequence S" of S' . Consider a circle Iz-I. I <e belonging to the domain D and a number 0(0<0< 1). Then by Theorem 8.19 we can find an integer Po~l such that when P~Po' the function fn, (z) = Hcn,z) has a (w) 2-circle in the circle IZ-I. I<e, for every value wEe, except at most the values w which belong to two spherical circles Iw , Wj I ~o (j = 1, 2). Consequently by Lemma 9. 3, when P~Po, the function Hz) has a generalized (w )2-circle in the circle r 0" qC n, ), for every value wEe, ex-
Apphcatwns of the Theory of Q .. -normal Famihes
359
cept at most the values w which belong to two spherical circles Iw,wjl~o(j=l,
2, ). The circles
r 0., £ ;cn)
(p~po)
are filling circles of the function f(z), in ex-
tended sense. By the same procedure, we can also obtain similar results from Theorems 8. 13, 8.14, 8.15 and 8.16. On the other hand, besides the filling circles and filling domains (in usual sense or in extended sense) of the function f(z) , we can also get Julia directions and Julia curves (in usual sense or in extended sense) of the function Hz) ,'provided that we choose suitablely the sequence of constants C n (n
= 1 ,2, .•• ) satisfying (9.3).
In fact, it is easy to see that for Julia direc-
tions, it is sufficient that C n(n= 1,2, ••• ) is a sequence of positive numbers satisfying the condition (9.3).
On the other hand, consider a curve z=qJ(t)(o~t<
+00), where qJ(t) is a continuous complex valued function for O~t<+oo, such that when t increases from 0 to +00, IqJ(t) I increases from 0 to +00. Then it is easy to see that for JUlia curves, it is sufficient to choose C n=qJ(tn)(n = 1, 2, ••• ) such that the condition (9.3) is satisfied (see Chapter 2, section 2. I).
9. 3. LIMITATION OF THE MODULUS OF MEROM ORPHIC FUNCTIONS We are going to give some generalizations of Schottky's theorem. A) Case of holomorphic functions
Lemma 9. 5. m ~ 0 being an integer, let $T be a Qm-normal family of holomorphic functions in a domain D. Suppose that there is a set ECD satisfying the following conditions: 1 0 E has the property W m with respect to D. 2 0 At each point of E the family $T is uniformly bounded. Then the family $T is uniformly bounded on each bounded closed set crCD.
Proof. Consider a bounded closed set crCD. Assume, on the contrary, that the family $T is not uniformly bounded on cr. Then to each positive integer n corresponds a function fn(z) of the family $T, such that
max If.(z) ,Eu
I>
n.
(9. 10)
Normal Familtes of Meromorphtc Functions
360
From the sequence S: fn (z) (n = 1,2, ... ) we can extract a subsequence S' : fn, (z) (k = 1,2, ••• ) which is a Cm-sequence in D. Let e be the set of nonCo-points of the sequence S' in D. By Theorem 8. 2, e does not have the property W m with respect to D. Since e ' Ce, the set Do=D-e is a domain, by a result obtained in section 8. 5. In Do, the sequence S' is a Co-sequence. By the condition 1°, there is a point Zo E E
nDo.
By the condition 2°, the sequence S' is bounded at zoo So
the limit function f(z) of this sequence in Do is a holomorphic function. So we can apply Theorem 9. 1 to the sequence S' and conclude that it converges locally uniformly to a holomorphic function in D. But this is in contradiction with (9.
10), and the Lemma 9.5 is proved.
Theorem 9.7. Given a domain D, an integer m?:O, a condition K of Qm_ normality, a set E CD having the property W m with res pect to D, a positive function M (z) defined on E and a bounded closed set aE D, we can find a positive number A (D ,m ,K ,E ,M (z) ,a) such that if f(z) is a holomorphic function in D satisfying the conditions:
°
1 f(z) satisfies the condition K in D, 2 0 1Hz) I~M(z) for zEE; then If(z) I ~ A (D ,m ,K ,E ,M(z) ,cr) for z E cr.
Proof. Let .J7 be the family of all holomorphic functions f(z) in D satisfying the conditions 1° and 2°. This family .J7 satisfies the conditions of Lemma 9. 5, hence .J7 is uniformly bounded on the set a, and accordingly the number A exists. In the particular case m = 0, the set E may be taken to be a single point of the domain D. So we have the following corollary:
Corollary 9. 2. Given a bomain D, a condition K of Qo-normality, a point Zo ED, a positive number M and a dounded closed set aCD, we can find a positive number A (D, K ,zo, M ,a) such that if Hz) is a holomorphic function in D satisfying the conditions:
°
1 f(z) satisfies the condition K in D,
Apphcatwns of the Theory of Q,.-normal Families
361
then
If(z)
1
~ A (D ,K ,zo,M ,0') for z
E
O'.
This corllary implies Schottky's theorem:
Corollary 9. 3. Let f(z) be a holomorphic function in the circle 1 z 1 <1 satisfying the following conditions: l°f(z) does not take the values 0 and 1 in the circle Izl
'I f
(z )
1~ a(M ,r ) ,
where a(M ,r) is a positive number depending only on M and r.
Proof. It is sufficient to define a(M ,r) = A (D ,K ,zo,M ,0'), where D is the circle Iz 1 <1, K is the condition of omitting to take the values 0,1,=,
zo=O, o is the circle Izl~r.
Lemma 9. 6.
m~l and v~O being two integers, let $T be a Qm-normal of
order v at most family of holomorphic functions in a domain D. Suppose that there is a set ECD satisfying the following conditions: 1° E has the property Wm-1,v+l with respect to D. (see Definition 9.1) 2° At each point of E the family $T is uniformly bounded. Then the family $T is uniformly bounded on each bounded closed set oCD.
Proof. Consider a bounded closed set oCD. Assume, on the contrary ,
362
Normal
Fam~hes
of Meromorplnc
Funct~ons
that the family .57 is not uniformly bounded on a. Then as in the proof of Lemma 9. 5, we get a sequence S:in (z)(n = 1, 2, •.• ) of functions of .57, satisfying the condit ion (9. 10). From S we can extract a subsequence Sf :in, (z) (k = 1 , 2, ••• ) which is a Cm-sequence in D and has at most v nonCm_l-points in D. Denoting by e the set of the nonCm_l-points of Sf in D, then Sf is a Cm_I-sequence in the domain DI =D-e. By the condition 1°, the set E (m-I) consists of at least
v+ 1 points,
r: Iz-zol
respect to D I . On the other hand, Sf considered as a family is Qm_I-normal in D I . Moreover by the condition 2°, at each point of E r , Sf is uniformly bounded. Consequently by Lemma 9. 5, Sf is uniformly bounded on each bounded closed set belonging to D I. Next by the maximum modulus theorem, we see that Sf is uniformly bounded on each bounded closed set aCD. But this is incompatible with (9.10), so Lemma 9.6 is proved.
Theorem 9.8. Given a domain D, two integers
m~l and v~O, a condi-
tion K of Qm-normality of order v at most, a set E CD having the property Wm_I,v+1 with respect to D, a positive function M(z) defined on E and a bounded closed set aCD, we can find a positive number A (D, m, v, K ,E, M (z), a) such that if iCz) is a holomorphic function in D satisfying the conditions: 1 ° iCz) satisfies the condition K in D, 2°lf(z) I~M(z) for zEE; then If(z) I ~ A
CD ,m ,v,K ,E ,M(z) ,u)
for z
E
u.
Proof. Let .57 be the family of all holomorphic functions fez) in D satisfying the conditions 1° and 2°. This family .57 satisfies the conditions of Lemma 9. 6, hence .57 is uniformly bounded on the set a, and accordingly the number A exists. In Lemmas 9. 5 and 9. 6 the set E is the same for all functions of the family .57. Now we give some extensions of these Lemmas, by making E variable with the functions of.57. First we give some definitions.
Apphcatwns of the Theory of Qm-normai Famthes
363
Definition 9. 3. Let En Cn = 1, 2, ••. ) and E be sets of points of the complex plane C. We say that E is a limiting set of the sequence E nCn=1,2,···), if for any point Zo E E, any positive number e and any positive integer N, we can find an integer n~N such that the set En
n ( Iz-zo I <e) is nonempty.
Definition 9. 4. Let D be a domain and (E) a collection of sets of points of D. Let m~O be an integer. We say that (E) is a (m)-collection with respect to D, if every sequence of sets En (n = 1,2,,,,) of (E) has a limiting set E CD having the property W m with respect to D. In particular, let
0
be a bounded closed set of points of D. If each point of
0
is considered as a set E, then the collection (E) is a (O)-collection with respect to D.
Definition 9. 5. Let D be a domain and (E) a collection of sets of points of D. Let m~O and v~l be two integers. We say that (E) is a (m ,v)-collection with respect to D, if every sequence of sets En (n = 1,2,,,,) of (E) has a limiting set ECD having the property W m " with respect to D.
Lemma 9. 7. m ~ 0 being an integer, let $T be a Qm-normal family of holomorphic functions in a domain D. Suppose that there are a (m )-collection (E) with respect to D and a positive number M satisfying the following condition :For each function fCz) E $T there is a set E =E (f) of the collection (E) such that If(z)
I~ M
for z
E E(j).
(9. ll)
Then the family $T is uniformly bounded on each bounded closed set aCD. Pr oof. As in the proof of Lemma 9. 5, assume that $T is not uniformly bounded on a bounded closed set oCD. Then there is a sequence of functions S: fn (z) (n = 1,2,,,,) of $T such that (9. 10) holds for each n. From S we can extract a subsequence S' : fn, (z) (k = 1 ,2, ... ) which is a Cm-sequence in D. As in the proof of Lemma 9.5, S' is a Co-sequence in a domain Do=D-e, where e does not have the property W m with respect to D. Consider the corresponding sequence of sets E. =E Ctn,) (k = 1,2,,,,) of the collection (E) , which has a limiting
364
Normal Famtltes of Meromorphic Functions
set E'CD having the property Wm with respect to D. There is a point zoE DonE'. Let
r : IZ-Zo I
be a circle such that
f:
IZ-Zo I~r belongs to Do. By Definition 9.
3, for any positive integer K, there is a positive integer k ~K such that the set E. n
r
is non-empty. This shows that in Do the limit function of S' is a holomor-
phic function. As in the proof of Lemma 9. 5, this leads to a contradiction with
(9.10).
Theorem 9. 9. Given a domain D, an integer
m~O, a condition K of Qm-
normality, a (m)-collectin {E} with respect to D, a positive number M and a bounded closed set oCD, we can find a positive number A(D,m,K,{E},M,o) such that if f(z) is a holomorphic function in D satisfying the conditions: 1° f(z) satisfies the condition K in D, 2° there is a set E of the collection {E} such that If(z) I~M for z E E ; then If(z)1 ~A(D,m,K,{E},M,iT) for zEiT.
To prove Theorem 9. 9, we consider the family
sr of all
holomorphic func-
tions in the domain D satisfying the conditions 1° and 2° and then apply Lemma
9.7.
Corollary 9. 4. Given a domain D, a condition K of Qo-normality, a bounded closed set eCD, a positive number M and a bounded closed set oCD, we can find a positive number A (D ,K ,e ,M ,0) such that if f(z) is a holomorphic function in D satisfying the conditions: 1° f(z) satisfies the condition K in D, 2°min If(z) I~M, zEe
then If(z)
Lemma 9. 8.
I ~ A (D
,K ,e,M ,iT) for zEiT.
m~1 and v~O being two integers, let
sr be a Qm-normal of
order v at most family of holomorphic functions in a domain D. Suppose that there is a (m-l, v+ 1 )-collection {E} with respect to D and a positive number M
Fam~lies
Apphcatwns of the Theory of Qm-normal
365
satisfying the following condition :For each function fez) E.7 there is a set E = E (0 of the collection {E) such that (9. 11) holds. Then the family .7 is uniformly bounded on each bounded closed set oeD.
Proof. Assume that .7 is not uniformly bounded on a bounded closed set oeD. Then there is a sequence of functions S:fn (z)(n = 1, 2, ••• ) of .7 such that (9. 10) holds for each n. From S we can extract a subsequence S' : fn,. (z)(k = 1, 2, .•. ) which is a Cm-sequence and has at most v nonCm_l-points in D. Let e be the set of nonCm_l-points of S' in D. Then in the domain DI =D-e, S' is a Cm_I-sequence. Consider the corresponding sequence of sets E. =E Un,)(k = 1, 2, ••• ) of the collection {E}, which has a limiting set EeD having the property Wm_I,v+1 with respect to D. Accordingly the set E(m-1J consists of at least v + 1 points. There is then a point zoEE(m-IJnD I. Let
r:
Iz-zol
T hen the set En r has the property W m-I with respect to
r.
On the other hand,
let el be the set of nonCo-points of S' in D I. Then el does not have the property W m-I with respect to D I. Consequently there is a point Zl E E n (DI-el)' S' is a Co-sequence in the domain D2=DI-el' Let
r I: IZ-Zl I
IZ-ZI I ~rl belongs to D 2· Since E is a limiting set of the sequence of sets E. (k = 1,2,,,,), hence by Definition 9. 3, for any positive integer K, we can find an integer k ~K such that the set E. n
r I is
non-empty. Consequently by the condi-
tion (9. 11), we see that the limit function of S' in D2 is a holomorphic function. By Theorem 9. 1, S' converges locally uniformly to a holomorphic function in D. This is incompatible with (9.10). Theorem 9.10. Given a domain D, two integers m~1 and v~O, a condition K of Qm-normality of order vat most, a (m-l ,v+ I)-collection {E) with respect to D, a positive number M and a bounded closed set oeD, we can find a positive number A (D,m ,v,K, {E) ,M ,0) such that if fez) is a holomorphic function in D satisfying the conditions: 1 0 fCz) satisfies the condition K in D, 2 0 there is a set E of the collection {E} such that IfCz) I ~M for z E E ; then
If(z)1
~A(D,m,v,K,{E},M,IT)
for
zEIT.
366
Normal Families of Meromorphic
Funct~ons
This theorem is deduced from Lemma 9. 8 in the usual way.
Corollary 9.5. Given a domain D, an integer
v~l, a condition K of QI-
normality of order vat most, a bounded closed set eCD, two positive numbers
5, M and a bounded closed set oCD, we can find a positive number A(D,v,K,e, 5,M ,0) such that if Hz) is a holomorphic function in D satisfying the conditions:
°
1 Hz) satisfies the condition K in D, 2 0 there exist v+1 points zj Ee(j=1,2,"',v+I) such that
(9. 12) and that
then If(z)1 ~A(D,v,K,e,o,M,u) for z E u.
In this Corollary it is tacitly assumed that there exist v+ 1 points
Zj
E e (j = 1,
2,···,v+I) satisfying the condition (9.12). To deduce this corollary from Theorem 9,10, it is sufficient to define a (0, v+ I)-collection (E) with respect to D, by choosing the sets E to be the systems of points
Zj
E e (j= 1,2, ••• , v+ I) satisfy-
ing the condition (9.12). (see Lemma 5.10 and its proof) B)Case of merom orphic functions In the case of meromorphic functions we can not expect to obtain an inequality of the form If(z)
1
~ A for z
E u,
as in the case of holomorphic functions, because the function f (z) may have poles on the set o. However we shall see that such an inequality can still be found, provided that
Z
is assumed not belonging to certain circles.
Lemma 9. 9. m ~ 0 being an integer, let .7 be a Qm-normal family of
Appltcat~ons
of the Theory of Qm-normal Families
367
meromorphic functions in a domain D. Suppose that there is a set ECD satisfying the following conditions: 1 0 E has the property W m with respect to D. 2 0 At each point of E the family sr is uniformly bounded. T hen for any bounded closed set aCD and any positive number 6, we can find a positive number o.(a,6) such that for each function f(z)Esr, the set
{zlz E
u,lf(z)1 >a(u,o)}
can be contained in a finite number of circles of which the sum of the radii is less than 6.
Proof. Let a be a bounded closed set of points of D and 6 a positive number such that a can not be contained in a finite number of circles of which the sum of the radii is less than 6. Suppose, on the contrary, that there does not exist a positive number o.(a,6) having the required property. Then to each positive integer n corresponds a function fn (z) E sr such that the set Ii. =
{z
Iz E
u, If'(z)
I> n}
(9. 13)
cannot be contained in a finite number of circles of which the sum of the radii is less than
o.
From the sequence S: fn (z) (n = 1,2, ••. ) we can extract a subse-
quence S' : f n• (z) (k = 1,2, ••• ) which is a Cm-sequence in D. Let e be the set of nonCo-points of S' in D. In the domain Do=D-e, S' is a Co-sequence. As in the proof of Lemma 9. 5, we see that the limit function Hz) (with respect to the spherical distance) of S' can not be the constant
=,
and hence Hz) is a mero-
morphic function in Do· Now distinguish two cases: 1) The set ane is empty. Then aCD o· On a the function Hz) has at most a
finite num ber of poles
Zj
(j = 1 ,2, .•. ,h). Consider the circles
r j: IZ-Zj I <0/ (h + 1)
h
(j = 1 ,2, "', h) and the set a1 = a- U r
j'
Evidently a1 is bounded closed and the
j=l
function Hz) is finite on a1' Hence Hz) is bounded on a1' By Theorem 1. 2 and the finite covering theorem, S' converges uniformly (with respect to the spherical distance) to Hz) on a1' Then it is easy to see that there is a positive integer ko such that the sequence fn• (z) (k ~k 0) is uniformly bounded on a1' But this is incompatible with the fact that the set
f.ln
n a1 is non-empty for each n.
Normal Families of Meromorphic Functions
368
2) The set an e is non-empty. By Theorem 8. 1, the set e is at most enu-
merable. So to each point
~
E e corresponds a positive integer n
(0
such that n
(~1):;;i:n(~2) for ~1:;;i:~2' Now with each point ~Eane associate the circle
I <0/2
n (tHl.
r t:
Iz-~
Then since the set ane is obviously bounded closed, there are a fi-
nite number of points ~j(j=1,2, .. ·,p) of ane, such that
,
(J'neCUI~. j=l
Note that the sum of the radii of
J
r t, (j= 1,2,'"
,p) is
The set
,
(J'. = (J'- U I ~ j=l
J
is non-empty and is bounded closed. Moreover a. CD o• Since aCa.
• t), U (Ur j=I
j
the set a. can not be contained in a finite number of circles of which the sum of the radii is less than 0/2. Setting 11.'
then Iln = Iln •
= {zlz E (J'.,lf.(z)1 >n},
• r ,,) and it follows that U(U
Iln'
(9. 14)
can not be contained in a finite
j~l
number of circles of which the sum of the radii is less than 0/2. Since a. CD o , we can carryon the reasoning as in case 1), in replacing a by a. , 0 by 0/2 and (9. 13) by (9. 14), and get again a contradiction. This completes the proof of Lemma 9.9. Theorem 9. 11. Given a domain D, an integer m;;:?;O, a condition K of Qmnormality, a set ECD having the property Wm with respect to D, a positive function M (z) defined on E, a bounded closed set aCD and a positive number 0, we can find a positive number A(D,m,K ,E,M(z),a,o) such that if f(z) is a meromorphic function in D satisfying the conditions:
369
Apphcations of the Theory of Q .. -normal Families 1 ° Hz) satisfies the condition K in D, 2 0 1Hz) I~M(z) for zEE; then the set
{z Iz E O',lf(z) I
> A (D,m,K,E ,M(z) ,O',o)}
can be contained in a finite number of circles of which the sum of the radii is less than
o. To prove this theorem, it is sufficient to consider the family $T of all mero-
morphic functions in D satisfying the conditions 1° and 2°, and to apply Lemma 9.9.
Lemma 9. 10.
m~1 and v~O being two integers, let $T be a Qm-normal
of order v at most family of merom orphic functions in a domain D. Suppose that there is a set ECD satisfying the following conditions: 1° E has the property Wm_I.,+1 with respect to D. 2° At each point of E the family $T is uniformly bounded. Then for any bounded closed set aCD and any positive number 0, we can find a positive number o.(a,o) such that for each function Hz) E $T, the set
{z Iz E O',lf(z) I
> a(O',o)}
can be contained in a finite number of circles of which the sum of the radii is less than
o. Proof. Let aE D be a bounded closed set and 0 a positive number such that
a can not be contained in a finite number of circles of which the sum of the radii is less than
o.
Suppoes that we can not find a positive number o.(a,o) having the
required property. Then there is a sequence S:in (z) E $T (n = 1, 2, .. ·) such that for each positive integer n, the set Iln defined by (9. 13) can not be contained in a finite number of circles of which the sum of the radii is less then
o.
From S we
can extract a subsequence S' :in, (z) (k = 1,2, .. ·) which is a Cm-sequence in D and has at most v nonCm_l-points in D. Let e be the set of nonCm_l-points of S' in D. In the domain DI =D-e, S' is a Cm_I-sequence, and hence S' considered as a family, is Qm_I-normal in D I. On the other hand, there is a point zoEE Cm -l)nD I. Let
370
Normal
Fam~hes
of Meromorplnc Functwns
r : IZ-Zo I
1•
Besides at each point of E I , S' is uniformly bounded.
Now distinguish two cases: 1) oCD I • In this case, by Lemma 9.9, we can find a positive number al
such that for each function fn, (z) E S' , the set
{z Iz E
iT,
If., (z) I > all
can be contained in a finite number of circles of which the sum of the radii is less than b. This is incompatible with the fact that the set
can not be contained in a finite number of circles of which the sum of the radii is less than b, when nk>al. h
2) The set one is non-empty. Let e=(zpz2,···,Zh) and ol=o-Ur p where ;-1
r; is the circle r;: Iz-z; I
+ 1).
h
Since oCol U ( U r j=l
;>,
01
COl is a bounded
closed set and can not be contained in a finite number of circles of which the sum of the radii is less than b/2. Again by Lemma 9. 9, we can find a positive number a2 such that for each function fn, (z) E S' , the set
can be contained in a finite number of circles of which the sum of the radii is less than b/2. It follows that the set
can be contained in a finite number of circles of which the sum of the radii is less than b. This is again incompatible with the fact mentioned above. The proof of Lemma 9.10 is now complete. From Lemma 9. lOwe deduce in the usual way, the following theorem:
Apphcatwns of the Theory of Q .. -normal Families
Theorem 9. 12. Given a domain D, two integers
371
m~1 and v~O, a condi-
tion K of Qm-normality of order v at most, a set E CD having the property W m-l,v+l with respect to D, a positive function M (z) defined on E, a bounded closed set oCD and a positive number 0, we can find a positive number A (D ,m , v,K ,E ,M (z) ,0,0) such that if f(z) is a meromorphic function in D satisfying the conditions:
°
1 Hz) satisfies the condition K in D, 2 0 1Hz) I~M(z) for zEE; then the set
{zlz E u,lf(z)1 >A(D,m,v,K,E,M(z),u,6)} can be contained in a finite number of circles of which the sum of the radii is less than
o. Lemma 9. 11.
m~O being an integer, let
.iT be a Qm-normal family of
merom orphic functions in a domain D. Suppose that there are a (m )-collection {E} with respect to D and a positive number M satisfying the following condition: For each function f (z) E.iT there is a set E = E (f) of the collection {E} such that
If(z)
I~ M
for z E E(f).
(9. 15)
Then for any bounded closed set oCD and any positive number 0, we can find a positive number u(o,o) such that for each function f(z) E.iT, the set
{z
Iz E
u, If(z)
I>
a(u,6)}
can be contained in a finite number of circles of which the sum of the radii is less than
o. Proof. Let
oCD be a bounded closed set and 0 a positive number such that
o can not be contained in a finite number of circles of which the sum of the radii is less than o. Suppose that we can not find a positive number u(o,o) having the required property. Then there is a sequence S:fn(z)E.iT(n=I,2,·") such that
Normal
372
Fam~hes
of Meromorplnc Functwns
for each positive integer n, the set Iln defined by (9. 13) can not be contained in a finite number of circles of which the sum of the radii is less than
o.
From S we
can extract a subsequence Sf :fn, (z)(k = 1, 2, ••• ) which is a Cm-sequence in D. In the domain Do=D-e, Sf is a Co-sequence, where e is the set of nonCo-points of Sf in D. Consider the corresponding sequence of sets Ek =E (fn,)(k = 1, 2, ... ) of
the collection {E}, which has a limiting set E'CD having the property Wm with respect to D. There is a point zoEDonE'. Let
f: /Z-Zo /:::;;;r
r:
/z-zo/
belongs to Do. By Definition 9. 3, for any positive integer K, there
is a positive integer k ~K such that the set Ek n
r is non-empty. This and the
condition (9. 15) imply that the limit function Hz) ( with respect to the spherical distance) of Sf in Do is not the constant
00,
and hence Hz) is a meromorphic
function in Do. The remaining part of the proof is the same as that of Lemma 9.
9. From Lemma 9.11 we deduce the following theorem: Theorem 9. 13. Given a domain D, an integer m~O, a condition K of Qmnormality, a (m )-collection {E) with respect to D, a positive number M, a bounded closed set aCD and a positive number 0, we can find a positive number A (D ,m ,K ,{E) ,M ,0,0) such that if fez) is a meromorphic function in D satisfying the conditions: 1 ° Hz) satisfies the condition K in D, 2° there is a set E of the collection {E} such that / Hz) /:::;;;M for z E E; then the set
{z/z E /T,/f(z)/ >A(D,m,K,{E},M,/T,d)} can be contained in a finite number of circles of which the sum of the radii is less than
o. Corollary 9. 6. Given a domain D, a condition K of Qo-normality, a
bounded closed set eCD, a positive number M , a bounded closed set aCD and a positive number 0, we can find a positive number A(D,K ,e,M ,0,0) such that if Hz) is a merom orphic function in D satisfying the conditions: 1 ° fez) satisfies the condition K in D, 2°min / Hz) /:::;;;M ; ,Ee
Apphcatwns of the Theory of Q .. -normal Famtltes
373
then the set {zlz
E IT,lf(z)1 >A(D,K,e,M,IT,o)}
can be contained in a finite number of circles of which the sum of the radii is less than O. To deduce this corollary from Theorem 9.13, it is sufficient to define a (0)collection {E} with respect to D, by considering each point of the set e to be a set E of the collection {E}.
Corollary 9.7. Given a domain D, a number h(O
and that Hz) does not take the values a;Cj= 1, 2, 3)in D, 2°min 1Hz) I~M; zEe
then the set {zlz
E IT,lf(z)1 >AI(D,h,e,M,IT,o)}
can be contained in a finite number of circles of which the sum of the radii is less than
o. This corollary can be deduced from Corollary 9. 6. It is sufficient to define Al (D ,h ,e,M ,IT,o) = A (D ,K ,e,M ,IT,o),
where K is the condition 1° in Corollary 9.7. K is a condition of Qo-normality by Theorem 3.9.
Lemma 9. 12. m?-1 and v?-O being two integers, Let .'3T be a Qm-normal of order vat most family of meromorphic functions in a domain D. Suppose that
374
Normal Families of Meromorphw Functwns
there are a (m-l ,v+ I)-collection {E} with respect to D and a positive number M safisfying the following condition :For each function f(z) E.57 there is a set E =E (f) of the collection {E} such that
If(z)
I~ M
for z E E(j).
(9. 16)
Then for any bounded closed set aCD and any positive number b, we can find a positive number u(a,b) such that for each function f(z) E.57, the set
{z
Iz E
(T, If(z)
I > a«(T,o)}
can be contained in a finite number of circles of which the sum of the radii is less than b.
Proof. Let aCD be a bounded closed set and b a positive number such that a can not be contained in a finite number of circles of which the sum of the radii is less than b. Suppose that we can not find a positive number u(a,b) having the required property. Then there is a sequence S dn (z) E.57 (n = 1, 2,,,') such that for each positive integer n, the set Iln defined by (9.13) can not be contained in a finite number of circles of which the sum of the radii is less than b. From S we can extract a subsequence S' :f.. (z) (k = 1,2,,,') which is a Cm-sequence in D and has at most v nonCm_l-points in D. Denoting by el the set of nonCm_l-points of S' in D. Then S' is a Cm_I-SeqUence in the domain DI =D-ei' Next let e2 be the set of nonCo-points of S' in Di' Then e2 does not have the property W m_1 with respect to D I , and S' is a Co-sequence in the domain D2=DI-e2' Define e=el Ue2' then D 2=D-e. Consider the corresponding sequence of sets Ek =E (fn.) (k = 1,2,,,') of the collection {E}, which has a limiting set ECD having the property Wm-I,v+1 with respect to D. There is then a point Zo E E(m-I) n D I. Let
r : IZ-Zo I
be a circle
belonging to D I. Then the set En r has the property W m_1 with respect to hence also with respect to D I. Accordingly there is a point Zl E E n D 2. Let
r , r I: I
Z-ZI I
r I is
non-
empty. This implies, by the condition (9.16), the limit function f(z) (with re-
375
Appltcatwnsof tlte Tlteory of Qm-normal Famtltes
spect to the spherical distance) of S' in D2 is not the constant
00
and hence fCz)
is a meromorphic function in D 2. The remaining part of the proof is the same as that of Lemma 9. 9. From Lemma 9. 12 we deduce the following theorem:
Theorem 9. 14. Given a domain D, two integers m?-1 and v?-O, a condition K of Qm-normality of order vat most, a (m-l ,v+ I)-collection {E} with respect to D, a positive number M , a bounded closed set aCD and a positive number b, we can find a positive number A(D,m,v,K ,{E},M,a,b) such that iff(z) is a merom orphic function in D satisfying the conditions: 1 0 fCz) satisfies the condition K in D, 2 0 There is a set E of the collection {E} such that
If(z) I~M
for zE E ;
then the set {zlz E u,lf(z)1 >A(D,m,v,K,{E},M,u,6)} can be contained in a finite number of circles of which the sum of the radii is less than b.
Corollary 9.8. Given a domain D, an integer v;;?: 1 , a condition K of Q1normality of order vat most, a bounded closed set eCD and a positive number.." a positive number M ,a bounded closed set aCD and a positive number b, we can find a positive number A(D,v,K ,e,.."M ,a,b) such that if f(z) is a meromorphic function in D satisfying the conditions: 1 0 fCz) satisfies the condition K in D, 2 0 there exist v+l points zj Ee(j=1,2,···,v+I) such that
(9. 17) and that
then the set
376
Normal Families of
{zlz E 17,lf(z)1
Meromorph~c Funct~ons
>A(D,v,K,e,1/,M,I7,o)}
can be contained in a finite number of circles of which the sum of the radii is less than
o. In this corollary and the corollary below, it is tacitly assumed that there ex-
ist v+1 points zjEe(j=1,2,···,v+1) satisfying the condition (9.17). To deduce Corollary 9.8 from Theorem 9. 14, it is sufficient to define a (0, v+ 1 )-collection {E} with respect to D, by choosing the sets E to be the systems of points Zj E e (j = 1, 2, "', v+ 1) satisfying the condition (9. 17). (see the proof of Corollary 9.5)
Corollary 9.9. Given a domain D, an integer
v~l, a number h(O
1), a bounded closed set eCD and a positive number Tl' a positive number M, a
bounded closed set aCD and a positive number 0, we can find a positive number A, (D ,v,h ,e ,Tl,M
,a,o) such that if f(z) is a meromorphic function in D satisfy-
ing the conditions: 10there are three values a jEC(j=1,2,3) such that
and that for each j O:::;;;j:::;;;3) , the equation f(z) =aj has at most v distinct roots in
D, 2° there exist v+l points z j Ee(j=1,2,"',v+1) such that
Iz r Z ,. I ~ 1/
(1 :::;;; j, j' :::;;; v
+ 1, J oF j' )
and that If(z)I:::;;;M () = 1,2,"',v+ 1); then the set {zlz
E 17,lf(z)1
>A,(D,v,h,e,1/,M,I7,o)}
can be contained in a finite number of circles of which the sum of the radii is less
Apphcat~on8
of the Theory of Q.. -normal
Fam~l~e8
377
than b. T his corollary can be deduced from Corollary 9. 8. by defining Al (D .v.h.e .T].M .a.b) =A (D .v.K.e .T].M .a.b). where K is the condition stated in 1 0 of Corollary 9. 9. K is a condition of Q I-normality of order vat most. by Theorem 5. 4. Finally we prove some theorems on the limitation of the spherical derivative of a merom orphic function.
Lemma 9. 13. m ~ 0 being an integer. let $T be a Qm-normal family of meromorphic functions in a domain D. and let {E) be a (m )-collection with respect to D. Then we can find two positive numbers M and r such that for any function Hz) E $T and any set E of the collection {E). there is a point zoE E satisfying the condition:
a(z.n ~ M for zED
n (Iz-zo 1< r).
Proof. Assuming that such positive numbers M and r do not exist. then to each positive integer n correspond a function fn(z) E $T and a set En of the collection {E) such that there does not exist a point Zn E En satisfying the condition:
aCz.f.)~nforzED
1 n (lz-z.I<-)· n
(9. 18)
From the sequence S:fnCz)(n=I.2.···) we can extract a subsequence S' :fn,(Z) (k = 1. 2 ••.• ) which is a Cm-sequence in D .and hence is a Co-sequence in a domain Do = D-e. where e does not have the property W m with respect to D. The sequence of sets En, (k = 1.2 .... ) has a limiting set
ECD
having the property W m
with respect to D. Accordingly there is a point zoEEnDo. Since S' constitutes a Q o-normal family in Do. hence by Theorem 1. 6. we can find a circle belonging to Do and a number u>O such that for
a(z .f.)
~
Choosing a positive integer K such that
k~l.
a for z E I.
r : 1Z-Zo 1
we have
C9. 19)
Normal Famtltes of Meromorphtc Functions
378
(9. 20)
by Definition 9. 3, there is a positive integer k;;?K such that the set En.
< t)
is non-empty. Let
t;,.1
t;,. be
a point of this set. Then
t;,. E En. and
n( Z-Zo 1
the circle
1
1z-
to T Consequently by (9. 19) and (9.20), we have
J(z,f.,>
1 Iz-s-.I <-. n,
This is incompatible with the fact that for each n, there does not exist a point Zn EEnsatisfying (9.18).
Theorem 9. 15. Given a domain D, and integer m;;?O, a condition K of Qm-normality and a (m )-collection {E} with respect to D, we can find two positive numbers M=M(D,m,K,{E}) and r=r(D,m,K,{E}) having the following property:lf f(z) is a meromorphic function in D satisfying the condition K in D, then for any set E of the collection {E}, there is a point zoE E such that J(z,f) ~M for zED
n (Iz-zol
To prove this theorem ,we consider the family gr of all meromorphic functions in D satisfying the condition K in D, and then apply Lemma 9. 13 to gr and
{E}.
Lemma 9. 14. m;;?l and v;;?O being two integers, let gr be a Qm-normal of order vat most family of meromorphic functions in a domain D, and let {E} be a (m-l,v+l)-collection with respect to D. Then we can find two positive numbers M and r such that for any function f(z) E gr and any set E of the collection {E}, there is a point Zo E E satisfying the condition: J(z,f) ~M for zED
n (Iz-zol
Apphcattons of the Theory of Qm-normal Famtlies
379
Proof. Assuming that such positive numbers M and r do not exist, then to each positive integer n correspond a function fn (z) E $7 and a set En of the collection {E} such that there does not exist a point Zn E En satisfying the condition (9. 18). From the sequence S :fn (z) (n = 1,2, ••. ) we can extract a subsequence S'
:tn,
(z) (k = 1,2, ... ) which is a Cm-sequence in D and has at most v nonCm_l-points in D. S' is a Cm_I-sequence in the domain DI =D-el' where el is the set of nonC m_l points of S in D. Let e2 be the set of nonCo-points of S' in D I , then e2 does not have the property W m-I with respect to D I , and S' is a Co-sequence in the domain D2=DI-e2. The sequence of sets En, (k = 1,2, ••. ) has a limiting set ECD having the property W m-I.'+I with respect to D. As in the proof of Lemma 9.12, we see that there is a point ZI E En D 2. The remaining part of the proof is carried out in the same way as that of Lemma 9. 13. From Lemma 9. 14 we deduce the following theorem:
Theorem 9. 16. Given a domain D, two integers
m~1 and v~O, a condi-
tion K of Qm-normality of order vat most and a (m-l, v+ I)-collection {E} with respect to D, we can find two positive numbers M=M(D,m,v,K, (E}) and r=r (D, m, v, K , (E }) having the following property:lf Hz) is a meromorphic function in D satisfying the condition K in D, then for any set E of the collection {E}, there is a point zoEE such that J(z,f) ~M for zED
n (Iz-zol
Theorems 9.15 and 9.16 are generalizations of Theorem 3.4.
9. 4. LIMITATION OF THE NUMBER OF THE ZEROS OF MEROMORPHIC FUNCTIONS We first state the following lemma which is an immediate consequence of Theorem 5. 19:
Lemma 9. 15. Let
$7 be a Qo-normal family of meromorphic functions in
a domain D. Then for any bounded domain .... such that number 0
< e < 1,
I = .... +J. . CD and any
we can find a positive integer N ( .... ,e) having the following
property: If Hz) is a function of $7, then for every value wEe, we have
Normal
380
Fam~l~esof Meromorph~c
n(f,w,J)
Functwns
~N(J,e),
except at most the values w which belong to a spherical circle Iw,wo(f) I<e. Here and in what follows, we shall use the notation number of the roots of the equation f (z)
n(f,w,~)
= w in an open set
~
to denote the
CD, with due
count of order of multiplicity. From this lemma we deduce the following theorem:
Theorem 9.17. Given a domain D, a condition K of Qo-normality, a domain ~ such that teger
I =-" +a-"CD and a number O<e
N(D,K,~ ,e)
such that if f(z) is a meromorphic function in D satisfying the
condition K in D, then for every value wEe, we have n(f ,w ,.1)
~
N(D ,K ,.1 ,e),
except at most the values w which belong to a spherical circle Iw ,wo(f) 1<e. To prove this theorem ,it is sufficient to apply Lemma 9. 15 to the family $T of meromorphic functions in D and satisfying the condition K in D.
Lemma 9. 16.
m~O being an integer, let $T be a Qm-normal family of
meromorphic functions in a domain D, let {E} be a (m )-collection with respect to D and let O<e<1 be a number. Then we can find a positive integer N and a positive number r such that for any function f(z) E $T, and any set E of the collection {E}, there is a point zoEE such that for every value wEe, we have n(f,w,Q)~N,
Q=D
n (Iz-zol
except at most the values w which belong to a spherical circle Iw,wo(f,E) I<e.
Proof. Assuming that such Nand r do not exist, then to each positive integer n correspond a function fn (z) E $T and a set En of the collection {E} such that there does not exist a point Zn E En having the following property:For every value wEe, we have
n(f.,w,Q.)~n,
Q.=D
1 n (lz-z.1 <-), n
(9.21)
381
Appitcatwns of the Theory of Qm-normal F'am!lws except at most the values w which belong to a spherical circle Iw, w n I <e.
From the sequence S dn (z)(n = 1, 2, ••• ) we can extract a subsequence Sf dn, (z) (k = 1,2, .•. ) which is a Cm-sequence in D, and hence is a Co-sequence in a domain Do=D-e, where e does not have the property Wm with respect to D. The sequence of sets En, (k = 1,2, •.• ) has a limiting set
ECD having the property W m E nDo. Let r : IZ-Zo I
with respect to D. Accordingly there is a point Zo E circle such that
f: I Z-Zo I ~p belongs to Do. Since
Sf constitutes a Qo-normal
family in Do, by Lemma 9. 15, we can find a positive integer P such that for k?::: 1 we have
n(f •• ,w,T)~P
which holds for every value wEe, except at most the values w which .belong to a spherical circle Iw, w~ I <e. Choosing a positive integer K such that
by Definition 9. 3, there is a positive integer k?:::K such that the set En, <
f)
n( IZ-Zo I
is non-empty. Let 1;,. be a point of this set. Then 1;,. E En, and the circle c: I
z-I;,.I
r.
Hence for every value wEe, we have
n(f •• ,w,c)~P
Theorem 9. 18. Given a domain D, an integer m?:::O, a condition K of Qmnormality, a (m )-collection {E} with respect to D and a number O<e< 1, we can find a positive integer N =N (D ,m ,K ,{E} ,e) and a positive number r=r (D, m ,K , {E } ,e) having the following property: If f(z) is a meromorphic function in D satisfying the condition K in D, then for any set E of the collection {E}, there
Normal
382
Fam~lies
of
Meromorph~c
Functwns
is a point zoEE such that for every value wEe, we have n(f,w,Q)~N, Q=D n
(Iz-zol
except at most the values w which belong to a spherical circle Iw,wo(f,E) 1<£. To prove this theorem, it is sufficient to apply Lemma 9. 16 to the family $T of the meromorphic functions satisfying the condition K in D and to the collection {E} and the number E.
Lemma 9. 17.
m~1 and v~O being two integers, let $T be a Qm-normal
of order vat most family of meromorphic functions in a domain D, let {E} be a (m-l,v+ I)-collection with respect to D and let O<E<1 be a number. Then we can find a positive integer N and a positive number r such that for any function f (z) E $T and any set E of the collection {E}, there is a point Zo E E such that for every value wEe, we have n(f,w,Q)~N,
Q=D n (Iz-zol
except at most the values w which belong to a spherical circle Iw,woCf,E) I<E.
Proof. Assuming that such Nand r do not exist, then to each positive integer n correspond a function fn (z) E $T and a set En of the collection {E} such that there does not exist a point Zn E En having the property that (9.21) is true for every value wEe, except at most those in a spherical circle 1w, Wn 1<E. From the sequence S: fn (Z) (n = 1, 2, ••• ) we can extract a subsequence Sf
:in. (z)
(k = 1,2, •.. ) which is a Cm-sequence in D and has at most v nonCm_l-points in D.
Sf is a Cm_I-sequence in the domain DI
= D-el' where el is the set of nonC m_l-
points of Sf in D. Let e2 be the set of nonCo-points of Sf in D I , then e2 does not have the property W m-I with respect to DI and Sf is a Co-sequence in the domain D 2 =D I - e 2·
The sequence of sets En. (k
= 1,2, ••• ) has a limiting set ECD having the
property W m-l.v+1 with respect to D. As in the proof of Lemma 9. 12, we see that there is a point ZI E
E nD 2.
Since Sf constitutes a Qo-normal family in D 2, the re-
maining part of the proof is carried out in the same way as in the proof of Lemma 9.16.
Apphcat~ons
of the Theory of Qm-normal
Theorem 9. 19. Given a domain D, two integers
Fam~l~es
383
m~l, v~O, a condition
K of Qm-normality of order vat most, a Cm-1, v+ 1)-collection {E} with respect to D and a number 0<£<1, we can find a positive integer N=NCD,m,v,K, {E} ,e) and a positive number r=rCD,m ,v,K, {E} ,e) having the following property:lf f(z) is a meromorphic function in D satisfying the condition K in D, then for any set E of the collection {E}, there is a point Zo E E such that for every value wE C, we have n(f,w,Q)~N, Q=D
n Clz-zol
except at most the values w which belong to a spherical circle Iw,woCf,E) 1<£. As for Theorem 9.18, we deduce Theorem 9.19 from Lemma 9.17.
9.5. DOMAINS COVERED BY THE VALUES OF MEROMORPHIC FUNCTIONS As simple consequences of Theorems 8.11, 8.12, 8.15, 8.16 and Theorems 8. 19, 8. 20, we are going to prove the following theorems which in a certain sense, are extensions of theorems on the domains covered by the values of holomorphic or meromorphic functions proved in Chapters 2,3,5,6, and 7.
Definition 9. 6. Let Ej(j= 1, 2,3) be three sets of points of the extended complex plane
C.
We say that E j(j=1,2,3) are uniformly mutually disjoint, if
there is a number <5(0<<5<1) such that for j#j' (l~j ,j' ~3) and wEEp w' E Ej"
we have aways Iw,w' 1><5.
Theorem 9. 20. Let D be a domain,
v~O an integer and
0(1)
'Zj(l~j~v+
1) a system of points of D, satisfying the condition (r 1) with respect to D. (see Definition 8. 8). Let E j Cj = 1, 2,3) be three uniformly mutually disjoint sets of points of ~r (j =
C.
Then we can find a positive number r such that the circles
I j, 1Z-Zj 1
1,2, .•• ,v+ 1) belong to D and a positive number A, both depending only
on D, v,o(l) and E/j=1,2,3), having the following property ,If fCz) is a meromorphic function in D such that
384
Normal
Fam~ltes
of
Meromorph~c Funct~ons
then there is an j(1~j~3) such that for each point wE Epf(z) has at least v+ 1 distinct (w)!-circles in D.
Proof. Assume that such r and A do not exist. Taking a positive number p such that the circles Iz-zjl~p(j=1,2,···,v+D belong to D, then to each positive integer n corresponds a meromorphic function fn (z) in D such that (9.22)
and that there are three points w/ n)EE;Cj=1,2,3) such that for each jCl~j~ 3), fn (z) has at most v distinct (w/n) )!-circles in D. By Theorem 8. 20, the family S :fn (z) (n = 1,2, ••• ) is Q!-normal of order vat most in D. Next by Theorem 8.11, the family S satisfies the condition (M) (see Definition 8.9) with respect to the system of points
a(I)
:Zj(l~j~v+ D. This is incompatible with (9.22).
Theorem 9. 2l. Let D be a domain and
a(I):zj(j~D be a sequence of
points of D, satisfying the condition (r'!) with respect to D. (see Definition 8.8) Let E j(j=1,2,3) be three uniformly mutually disjoint sets of points of
C.
Then
we can find a positive integer p, a positive number r such that the circles T j: IzZj I~r (j = 1 ,2, ••. , p) belong to D and a positive number A, all depending only on D, a(I) and E j (j = 1 ,2,3), having the following property: If fCz) is a meromorphic function in D such that
then there is an j (l~j~3) such that for each point wE E j ,fez) has a (w )2-circle in D.
Proof. Assume that such p, r and A do not exist. Taking a sequence of positive numbers Pp (p = 1,2, ••• ) such that
Apphcat!onsof the Theory of Qm-normal Fam!lies lim p, 1--1+=
385
= 0
and that for each p (p:;?Ol) the circles F/p): IZ-Zj I ~pp (j = 1,2, ••• , p) belong to D, then to each positive integer p corresponds a meromorphic function fp (z) in D such that (9.23)
and that there are three points w/ P)EE j(j=1,2,3) such that for each j(1~j~3) fp(z) has no (w/P»2-circle in D. By Theorem 8.19, the family S:fp(z)(p= 1,2, ... ) is Ql-normal in D. Next by Theorem 8.12, we can find an integer po:;?ol such that the family S satisfies the condition (M) with respect to the system Zj(l ~j~po). Accordingly by Definition 8.9, we can find circles
C;:
Iz-zjl~rj(j=l,
2,,,, ,Po) belonging to D and a number 0.>0 such that for each p:;?ol, we have
N ow choosing a positive integer P such that (9.24)
then min (max J(z,fp) l~j:~P
zEr/1" ~
~
min (max J(z,fp» l~):;;;;;,o
zEr/l')
min (maxJ(z,fp»~a.
l~i~'o
zEG)
this is incompatible with (9.23) and (9.24). Similarly basing upon Theorems 8.15, 8.16 and Theorems 8.19, 8.20, we can prove the following two theorems:
Theorem 9.22. Let D be a domain, m:;?o2, v:;?oO two integers and Zj,j,"jm
(l~jl ~ v+
a(m):
1 , j2:;?o 1 , ... ,jm:;?ol) be a m-ple sequence of points of D, satisfy-
ing the condition (1 m) with respect to D. (see Definition 8.8) Let E j (j=1,2,3)
386
Normal Famthes of Meromorphtc Functwns
be three uniformly mutually disjoint sets of points of
C.
Then we can find a posi-
tive integer p, a positive number r such that the circles f ili, ...;,.: Iz-zili, ... imI:::;;;r (1:::;;; jl:::;;; v+ 1, 1 :::;;;j2:::;;;P , ••• , 1 :::;;;jm:::;;;P) belong to D and a positive number A, all depending only on D, m, v, oem) and E i (j=1,2,3), having the following property: If f(z) is a meromorphic function in D such that (maxa(z,f»~A,
mtn 1~jl~v+l.l~j2:s;;;,.···.l:S;;;j.. ~1
zEFj]jz
oj ..
then there is an jO:::;;;j:::;;;3) such that for each point wEEp f(z) has at least
v+ 1
distinct (w)m-circles in D.
Theorem 9. 23. Let D be a domain, 1,jz~1"",jm~1) (r~)
m~2 an integer and oem) :Zjli, ...;,. (jl~
be a m-ple sequence of points of D, satisfying the condition
with respect to D. (see Definition 8. 8). Let E j(j = 1,2,3) be three uniform-
ly mutually disjoint sets of points of
C.
Then we can find a positive integer p, a
positive number r such that the circles fili, ... jm: IZ-Zjli, ... jm I:::;;;r 0 :::;;;jl:::;;;P , 1 :::;;;jz:::;;;p , .•• , 1:::;;;jm:::;;;P) belong to D and a postive number A, all depending only on D, m, oem) and E j (j = 1, 2,3), having the following property:lf Hz) is a meromorphic function in D such that
then there is an jO:::;;;j:::;;;3) such that for each point wEEp f(z) has a (W)m+lcircle in D.
Lemma 9. 18. Given any integers
m~1 and n~1, we can find in the in-
terval I: O<x<1, a m-ple sequence of points oem) :Xjlj, ... j. O:::;;;jl:::;;;n ,j2~1, ••. ,jm~ 1) satisfying the condition (r m) with respect to the interval I, and a m-ple sequence of points
01 (m)
:Xjli, ... j. (jl~1 ,j2~1, ••• ,jm~l) satisfying the condition (r~)
with respect to the interval I. (see Definition 8.8, for I)
Proof. For the case m= 1, o(]) :xil O:::;;;jl:::;;;n) is a system of points. In this case, the existence of
0(J)
is evident. Now assume that for an integer m~1, we
can find in the interval I a m-ple sequence of points oem) :Xjlj, ... j. O:::;;;jl:::;;;n ,jz~1,
387
Appltcations of the Theory of Qm-normai Famihes
••• ,jm~l) satisfying the condition (r m) with respect to I- Let us define a (m 1)-ple sequence of points
a(m+l)
+
:Xi,i,"'i.+, (1~jl~n ,j2~1, ••• ,jm+I~1) of I as fol-
lows: To each point Xi,i,"'i. associate an increasing sequence of points X/i,i,···i.) O~ 1) of the interval O<X<Xi,J""i. such that
lim X
j-4+ oo
.(j,j,"'j.) )
=
X ..•••. )1)2
J.
and then define
then evidently
a(m+l)
satisfies the condition (r m+l) with respect to I-
On the other hand, for the case m = 1,
al ( I ) :
Xi, (jl ~ 1) is a sequence of
points of I- In this case, it is suficient to take an increasing sequence ~l) of points of I, such that the limit of
al (I)
al (I)
:xi, 01
belongs to I- Then evidently
al (I)
satisfies the condition (r'l) with respect to I- Next by mathematical induction, we prove as in the above, the existence of
Definition 9. 7. Let qJ(x)
al
(m) for each m~1.
(O~x~ 1) be a complex valued continuous
function in the interval O~x~ 1, such that for XI #X2 (O~xp X2~ 1) we have qJ (XI)#qJ(X2)' Then we call qJ(x)(O~x~1) a simple curve and denote it by C.
Lemma 9. 19. Let D be a domain and C :qJ(x) (O~x~1) be a simple curve belonging to D. Let m~l and n~l be two integers. Then the following two assertions are true: 1° If a(m):xi,i""i.(1~jl~n,j2~1,···,jm~1) is a m-ple sequence of points of the interval I:O<x
al
(m) :Xi,i,"'i. (jl~l ,jz~l, .•• ,jm~1) is a m-ple sequence of points of the
interval I: O<x
Normal Famtltes of Meromorp/nc Functwns
388
a system fo points, and the assertion lOis evidently true. Now assume that lOis true for an integer m~l. Consider a (m
+ 1 )-ple sequence
a(m+l)
:xJ,J,"'im+, (l~jl ~
n,j2~1, .. ·,jm+l~I) of points of I, satisfying the condition (r m+ 1 ) with respect
to I, and the corresponding (m
+ 1 )-ple sequence
i1(m+l)
:zJ,J,"'Jm +, =CP(XJ,i''''Jm +,) (l~
jl ~n ,h~l,'" ,jm+l~I) of points of C. Then by Definition 8.8 (for I), for each system jl' h, ... ,jm (1 ~jl ~n, j2~ 1, "', jm ~ 1), the sequence of points Xi,i,"·i.im+, (jm+l ~I) are distinct and the limit
exists and belongs to 1. Hence the sequence of points Zi,i,"·i.im+, =CP(Xi,i''''imim +,) (jm+l ~I) are distinct and the limit
exists ad belongs to C. Moreover by Definition 8. 8 (for 1), the m-ple sequence ~1,i2"".O~jl~n,j2~1, .. ·,jm~1) satisfies the condition (r m) with respect to 1.
Hence by assumption, the m-ple sequence CP(~i,i''''i) (1 ~jl ~n, j2~ 1, ... ,jm~ 1) satisfies the condition (r m) with respect to D. Consequently by Definition 8. 8, the (m
+ I)-pie sequence i1(m+lJ satisfies the condition
(r m+l) with respect to D.
To prove 2°, consider first the case m=l. In this case, a/l):xi,(jl~I) is a sequence of distinct points of I, such that
ltm x j =';
j]--1+=
I
exists and belongs to 1. Hence the sequence i1/lJ: Zi, = CP(Xi) (jl ~ 1) are distinct points of C, such that
exists and belongs to C. So in this case, 2° is true. As for 1°, we complete the proof of 2° by mathematical induction.
Theorem 9.24. Let D be a domain,
m~l, v~O two integers and C :cp(x)
Applicattons of the Theory of Q .. -normal Families (O~x~l)
389
a simple curve belonging to D. Let E j (j=1,2,3) be three uniformly
mutually disjoint sets of points of
C.
Then we can find a positive number A de-
pending only on D, m, v, C and E j (j = 1 ,2,3) having the following property: If f (z) is a meromorphic function in D such that
mtnJ(z ,f)
~
A,
(9.25)
zEC
then there is an j(1~i~3) such that for each point wEEp Hz) has at least v+l dis tinct (w )m-circles in D.
Proof. First of all, by Lemmas 9. 18 and 9. 19, we can find a m-ple sequence of points iJ(m): Zj,J, ••. J,. (1 ~jl ~ v+ 1, i2 ~ 1, ••• , im ~ 1) of C, satisfying the condition (r m) with respect to D. Next by Theorems 9.20 and 9. 22, we can find a finite number of terms \:.,(i=1,2,···,q) of iJ(m), a positive number r such that the circles ;f,:
IZ-\:.I I~r (i =
1,2, .•• ,q) belong to D and a positive number A, hav-
ing the following property: If fCz) is a meromorphic function in D such that
min (maxJ(z,f»
l~j~q
~
A,
(9.26)
zEI;
then there is an i(1~i~3) such that for each point wEEp Hz) has at least v+l distinct (w)m-circles in D. Since (9.25) implies (9.26), hence this positive number A has the required property. Similarly we can prove the following theorem:
Theorem 9.25. Let D be a domain,
m~l,v~O,k~1 three integers and
C :
minJ(z ,f)
~
A,
(9.27)
zEC
then one at least of the following two assertions holds: 1 0 For each point w of the circle
Iwi <M,
f (z) has at least v+ 1 distinct
390
Normal Families of Meromorplltc Functions
(w)m-circles in D. 2° For each point w of the domain
b< Iwi <M ,
f(O) (z) has at least v+ 1 dis-
tinct (w )m-circles in D.
Proof. Assume on the contrary, that there does not exist a positive number A having the required property. Then to each positive integer n corresponds a meromorphic function fn (z) in D, such that minJ(z,f.)
~n
(9.28)
zEC
and that there are two points w~ and w~ respectively of the circle Iw I<M and the domain
b< Iw I<M , for
which fn (z) has at most v distinct (w~ )m-circles in D and
fn (0) (z) has at most v distinct (w~)m-circles in D. By Theorem 8. 22, the family of functions fn (z) (n = 1,2, ••• ) is Qm-normal of order 2vat most in D. By Lemmas 9. 18 and 9. 19, we can find a m-ple sequence of points h~l,
'"
,jm~l)
a(m)
:Zi,i,"'jm
(1~jl~2v+ 1,
of the curve C, satisfying the condition (r m) with respect to D.
N ext by Theorems 8. 11 and 8. 15, we can find a finite number of points 1;" (j = 1 , 2,,,, ,q) of
a(m)
and circles
I,: Iz-1;" I~r, (j =
1 ,2, ••• ,q) belonging to D and a num-
ber a>O, such that for each n we have min (maxJ(z,f.»
l:e;;i~q
~
a.
Evidently (9.28) and (9.29) imply
n~a
and hence we get a contradiction.
As an example, consider the entire function fJ.(z)=e"',
where ",>0 is a constant. We have If~(z)
J(z,fJ.)
So for real y we have
(9.29)
zEf;
I
= 1 + IfJ.(z)12
Ale'" I
Apphcatwns of the Theory of Qm-normai Famihes
391
which can be as large as we please, provided that" is sufficiently large. Accordingly by Theorem 9. 21, for any integer m;> 1, f, (z) should have (w )m-circles (w
# 0,(0), when" is sufficiently large. We are going to show that this is really true. In fact, consider a circle T : 1z-iyo 1 <0, where Yo is real and 0< Til, Tll being given in Definition 8. 13 and a domain E : I/R < when
,,>/--] , ,,] =
1
w 1 1). Let us show that
(JogR +2rr) /0, the function f, (z) takes in T every value wEE.
Accordingly, by Definition 8. 13, for every wEE, the circle T is a (w) I-circle of f,(z). To prove this consider a point w=pe"EEO/R
">1.].
Let k be the integer such that 2k1r:::;; ),y 0-8
<
2(k
+ 1)n-,
and set
~=
-l(109 P
+ i(8 + 2k1r».
Then
1
1~-~Yol:::;; ;,,(iogR
<
1
;,,(iogR
+ 12k1r-().y0-8) I) + 2n-) < o.
On the other hand, we have f,(~)
=
e~
= pe;e = w.
This proves our assertion. Consider again the circle T, but now assume that 0
f; : 1z-iYi 1 :::;;0'
(j= 1, 2, ... ,1 2) belong to
r and are mutually
392
Normal F am~lies of Meromorp hic Functions
disjoint. By the assertion just proved, we see that we can find a number "z> 0 such that for ">"zand wEE, each of the circles r;(j=I,2,···,Iz)is a (w)l-circle of the function f),(z) in r. Consequently r is a (w)z-circle of f),(z).
In general, basing upon Definition 8. 13, it is easy to prove by mathematical induction, the following assertion: Given an integer m~l, a circle r: Iz-iyoI 1), we can find a number A>O such that for ,,>A and wEE, r is a (w)m-circle of the function fx(z).
APPENDIX A A GENERAL THEOREM ON HOLOMORPHIC FUNCTIONS IN THE UNIT CIRCLE
The purpose of this appendix is to prove by the method of Wiman-Valiron in a simplified form due to Macintyre [21, 22J, a general theorem on holomorphic functions in the unit circle and to deduce from it some consequences. For this we need to prove first a theorem on convex functions.
1. A THEOREM ON CONVEX FUNCTIONS
Definition 1. f(x) being an increasing convex function in an interval, we denote by f~ (x) and C (x) the righ t-hand derivative and the left hand derivative of f(x) respectively.
Definition 2. Let u>O and 0<1.<1 be two numbers. Let U (x) be a positive non-decreasing function for
x~O such that
fdX/U (x) converges. We de-
fine 2a
H a.I.,I' = 1 _ ),
+
H
2a
f
dx U (x)
and
a,l,U
a,I.,u(X) = ---X-1-),
393
+
r
dx
x U(x)'
394 The function <1>•. ,..u (X) is positive, decreasing and continuous for X>O and satisfies the conditions ltm(/)a.)..u (X) x~o
= + 00,
ltm (/)a.)..V (X) x~+=
= O.
We denote by X=\If •. ,..u(Y) the inverse function of Y=<1>.,,.,u(X). The function \If •. ".U (Y) is positive, decreasing and continuous for Y>O and satisfies the condi-
tions: LIm If'a.)..u(Y) y~+=
=
0,
l!mlf'a.)..u(Y) y~o
=+
00.
Theorem 1. Let m (t) be an increasing convex function for t
for t
Assume that there exists a number to
Under these conditions, there exist three numbers t j (j=1,2,3) such that 1) -H •. ,.,U:(;tl
2)m (t 2) -m (t 1 ) =m (t3) -m (t 2) =a; 3»,(t2-tl)
5)m'+ (t 2):(;U {m (t 2)}. For the proof of this theorem we need some lemmas.
Lemma 1. Let d n (n~O) be a non-increasing sequence of positive numbers such that L d n converges. Let 0<>.<1 be a number. Let V (y) be a positive n=O
non-decreasing function for
y~O such that
r
dy/V (y) converges. Assume that
395
Appendix A
Under these conditions. there exists an integer no~l such that
d. o
>).d.
1
0
-I>
(2)
d. 0 ~ V no().
Proof. We have
~
~
.13}· ~ )..13}·-1 + 0 =
1
~
).)'~I
V (n)
.~J).d'-I + 0-).) vfn)/·
Hence there is an integer no~l such that
d. o ~).d. 0 - I
+ 0 - ) . ) V1-no()
which togather with the inequality dno-I~dno' gives (2).
Lemma 2. Let
dn(O~n~N .N~l) be a non-increasing finite sequence of
positive numbers. Let A and V (y) have the same meaning as in Lemma 1. Assume that
Under these conditions. there exists an integer l~no~N such that
d. o
>).d.
0
-I'
d. 0
1 > -V no().
(3)
Appendix A
396
Proof. We have
N
N
/~}.
> do + }'.I!}' + (1
N
'~1d.
-
N
> ).'~1d'-1 + (1 =
N 1 ).)'~1 V (n)'
1
N
).)'~1
-
'~1{M'-1 + (1
-).)
V (n)
vfn))'
Hence there is an integer l~no~N such that
1
d.>M.1 +(1-).)v-( o 0 no )
which togather with the inequality dno-1~dno' gives (3).
Lemma 3. Let h (s) be an increasing convex function for a ~s
b -
a
~ It't-
1
2a
(a) 1 -
).
+
r
(a)
dx U (x)
Under these conditions, there exist three numbers Sj(j= 1,2,3) such that l)a~s1<s2<s3
2)h (S2) -h (S1) =h (S3) -h (S2) =a; 3)I-.(S2-S1)<S3-S2~S2-S1
;
4)h(s2)~a;
5)h't-(s2)~U {h(s2)}'
Proof. Distinguish two cases as follows:
(4)
397 Case 1. The function h(s) is unbounded for a~s
+ na
and put (n ~
0).
We have
(5)
d.>O, Ed.=b-a, R=O
Moreover by the convexity of the function h (s), the sequence d. is non-increasing. Define
1
V (y) = -u {It (a)
a
v (y) is positive non-decreasing for
r
(a)
+ ay}
y~O.
dx U (x)
=
(y ~ 0).
(6)
We have
r 0
dy V (y)'
(7)
The relations (4), (5) and (7) yield
Hence by Lemma 1, there exists an integer no~l such that
d"
> Nl,,-l'
1
d"
~ V (no)'
(8)
By means of the convexity of the function h (s) and (8), we see that the three
398
Appendix A
values Sl =Uno-I
,S2=Uno
,S3=Uno +1 satisfy the conditions 1) -5) in Lemma 3.
Case 2. The function h(s) is bounded for a~s
Then by the convexity of the function h(s) and (4), we get
h(b) -
h(a)
~
(b -
a)h,+ (a)
> 1 ~ ),> 2a.
Let P~2 be the integer such that
< 11 (b)
p
Let
un(O~n~P)
-
11 (a) ~ p
a
+ 1.
be the finite sequence of values of s defined by the equality h(u.)
=
h(a)
+ na.
Put N
=
P -
1,
d.
= U.+I -
U.
(O~n ~N).
We have
d.
> 0,
(9)
and from the inequality h' (a) ~ 11 (up) + "'" Up -
we have
h (a) ~ 11 (b) a "'" b -
h (up) ~ _ a _ Up "'" b - Up
Append~x
399
A
(0)
Moreover by the convexity of the function h(s), the finite sequence do is non-increasing, The relations (4), (7), (9) and (0) yield
,Eod. > 1 do-)" + N
r
dy V(y)'
Hence by Lemma 2, there exists an integer l~no~N such that
d" >)"d, o- ] I d, 0
> V1-no()'
(1)
Then, as in case 1, by means of the convexity of the function h (s) and (1), we see that the three values Sl =00,-1 ,S2=00,' S3=00,+1 satisfy the conditions 1)-5) in Lemma 3, Now let us come back to the proof of Theorem 1. Put H=Ha,A.U and distinguish two cases according to
m(-H)~O
or <0,
First assume that m(-H)~O, In this case, consider the increasing convex function met) for -H~t
H">-: 1 ~+r<_H)U(X)' ~ ::O--m'+(_H) 1-)" Hence by Lemma 3, there exist three numbers tj (j= 1, 2,3) satisfying the conditions 1)-5) in Theorem 1. Now assume that m(-H)
- to ~ t1> •• J..u{m(to)}
=
H 2a m(t o) 1-)"
+
The function m (t) being increasing, the inequalities m(-H)
m(to»O
r
dx
(t,)U(x)'
(12)
Append ~x A
400 imply
- H
< to.
(13)
Then by the convexity of the function m (t), we get
, ( ) >-: m (to) - m (-
m+
to::--
to
+H
H)
>
m (to) H
(14)
•
The inequalities (12) and (14) yield
-t> 1 ~+ o m't- (to) 1 - A
r
(',J
~
(15)
Vex)·
Consider the increasing convex function met) for to~t
Corollary 1. Let met) be an increasing convex function for t
for tO and 0<,,<1 be two numbers. Define
It a,A =
2a 1_ A
+ 2,
¢a,,(Y)
= 1+
(lI a"
l'~
1)2
(Y
>
0).
(16)
Assume that there exists a number to
Under these conditions, there exist three numbers tj(j = 1, 2,3) such that 1) -h"'A~t]
4)m (t2)~U; 5)m'+ (t2) ~l
+ {m (t2) }2.
401
Appendix A
Proof. Let U (x) be the function defined for U (x)
= 1 for 0:::;;; x < 1,
U (x)
x~O
=
X2
-
1)2
as follows:
for x ~ 1.
Then it is easy to see that
H o,.a.,U
=
,(X) =
t[J
ha ,).,
(h a ,).
a,).,L
X
(X ~ 1)
and
1J! a,).,V (Y)
=
(lt a ,).;- 1)2
1J! a,).,V (Y)
<
1
(Y
>
(0
(h a ,).
-
:::;;;
(Ita,). -
1)2)
1)2).
Hence 1J! a,).,£' (Y)
< 1fa,).(Y)
(Y
> 0).
Besides we have (x~O).
U(x):::;;;1+x 2
Consequently Corollary 1 follows from Theorem 1.
2. A GENERAL THEOREM ON HOLOMORPHIC FUNCTIONS IN
THE UNIT CIRCLE Definition 3. Hz) being a holomorphic function in the circle Iz I<1, we define
M(r,f)
=
max If(z) I'I~'
I
(0
1).
402
Appendix A
Theorem 2. Let f(z) be a holomorphic function in the circle Izl
+
M(ro,f) ~ 4e(S
If(O)
De-I:.".
Under these conditions, we can find two numbers p, N such that
e- b < p < 1, M = M(p,f)
> S,
1~N < 1
+
(log
M sY
and two holomorphic functions u«;,) ,v«;,) in the rectangle
(R): -
2
<
Re(t;)
1 < - 2'
0< im(t;)
<
471'
having the following properties: 10In(R), u(1;,) is univalent and Re{u«;,)}
(R,,): -
2
1 < Re(t;) < - 2'
w < im(t;)
In (R), we have
2° In (R), we have
1v (t;) 1< and
271'
+ 41
< w + 271'.
403
Appendix A 3 0 For n~1, we have
In this theorem, a, b ,c are absolute positive constants whose values will be written out explicitly.
Proof. Define
f 1 (z)
= A1 f
(z ), A
=
2 (8
+
If (0)
I ).
(18)
We have
(19)
Next define
(20) We have f2(0)
= 0,
M(r,f2) ~M(r,fl) -
M(ro,f2)
1,
> eH-t:;',o.
In what follows, we first study the function f 2 (z), then the function fl (z) and finally the function f(z). Study of the function f 2 (z). By a theorem of Hadamard, the function m(t)
=
{ogM(e',f2)
(t
<
0)
(21)
404
Appendtx A
is increasing and convex for t <0. Since f 2 (z) =zg (z), where g (z) is a holomorphic function in the circle
Iz I<1,
met)
=
t
we have
+ logM(e',g)
(t
<
0)
which 'implies that
m't-
(t) ~m'_ (t) ~ 1
<
0).
(22)
(to = logro).
(23)
(t
Moreover we have
m(to)
> 1 + _a to
Now let us choose
a
= 3e4 4 ,
A = _a_ a+l
(24)
and a
=
(h a .> -
1)2,
(25)
where h a •• is defined by (16). Then (23) becomes
where 1j.'Q,'(Y) is defined by (16). Applying Corollary 1 to met), (I.,/., we conclude that there exist three numbers t j (j=1,2,3) satisfying the conditions 1)5) in Corollary 1. Put p =
e'·.
(26)
By the condition 1) in Corollary 1, we have (27)
Appendtx A
405
with
b
Taking a point
Zo
of the circle
1Z 1
=
= p,
(28)
h a ,;..
such that
(29) and introducing the number
(30)
we are going to show that N is real and
(31)
In fact consider a circle
r : 1 Z-Zo 1 <<'>
in which f 2(z)#0 and let J(z) be a branch
of the logarithm of f2 (z), holomorphic in
r.
Then putting
we have
tN =
Re(tN) =
J~I(peiO»)
\ dB
O~Oo
,
LdiOg If2(peiO) 1)._0' -
Since the function log If 2(pe;')
1
0
has a maximum at 8=8 0 , hence Re(iN)=O and N
is real. On the other hand, we have
406
AppendlX A
which may be written in the form
Consequently if we define
then N = IlmQ(r). '-1P
Finally from the inequalities Q (r)::::::: logM(r,f2) -logM(p,f2) ~ log r _ log p
(p
Q(r) ::::::: logM(r,f2) - logM(p,f2) ~ logr-logp
(O
1) ,
we deduce (31). The inequalities (22), (31) and the condition 5) in Corollary 1 yield (32)
Consider the function
(33)
holomorphic in the half plane Re (-1:)
< -t
2•
Putting
407 (34) we are going to show that
lepCr) 1< e for - a' :::;;; Re(-.):::;;; u.
(35)
In fact, we have
Icp(-.) I:::;;; em(',+.J-m(',J-N.
(u = Re(-.».
Since (31) implies
if follows that if O
m(t z
+ u) -
m(t z ) - Nu = u(m(t z
+ u~
- m(tz) - N}
(m (tz + u~ - m(t z) _ N}:::;;; u( m(t z + u~ - m(t z) m(t z + u)-m(t z ) - uN = a - uN,
:::;;; u =
and if -0'
=
m (t z -
a' ) - m (t z )
+ a' N
=
a' N - a.
Consequently
IcP( -.) I :::;;; eo-uN 1cp(-.)I:::;;;eu'N-O
for 0 :::;;; R
e-. : :; ; u,
for-a' :::;;;Re-.:::;;;O.
_ N}
408
Appendtx A
Finally by the conditions 2) and 3) in Corollary 1, we have
(36)
and we get, by (24),
(J
(IN ~ a(1 -
a -
u' N -
a
~ a(~ (J
u' ) -
1)
<
a(1 -
).)
<
a( ~ -
1) = 1-
).
<
1
Hence we have (35). We have
IP
(37)
From (35) and (37), we can apply Schwarz's lemma to the function <:pCr)-1 and get
fOT
l-rl
~
(J.
In particular, in the circle
(C):I-rl<
(Jr::'
16 V 3 we have
IIP<-r) -
11
<
4 1 1 3(16)2 = 1216'
and from (33),
(38)
409 Appendix A
with
I1/1 C.) I <
1 1 12 16
(39)
in (C). Now consider the function
(40)
We see easily that
(41)
By Cauchy's inequality and (35). we have
u
Zlrp'
C.)
I<
u
e
Z
fOT -
~ ReCr) ~
u
Z·
Hence
Irp'
C.)
N
I<
~ Nu'
By (36).
Nu
>
).uI
N ~ ).a
=
a
aa
Consequently
Irp'
(-r) N
I<
4e a
a
+ 1 > Z·
=-.l-.l
12 16'
(42)
Append~x
410
A
It follows that we have
(43)
with
(44)
in (C). By the condition 2) in Corollary 1, we have
n;;;:l being an integer, we have, by Cauchy's inequalities,
From (34) and (42), we have
p' -
p = pee" -
1)
> PO',
NO'>
a
2'
hence
(45)
Study of the function fJ (z). By (19) and (20), we have
(46)
411
Appendtx A and by (29) and the condition 4) in Corollary 1, we have
On the other hand from (38),(39),(29) and (36), we have in the circle (C),
Hence in the circle (C), we have
fl(zOe') =f2(zOe'){l +b(-r)},!b(-r)!
2 <---;;.
(49)
e"2
It follows from (38), (47) and (49), that
Consequently if we write
(50)
then we have
d(-r)
=
b(-r) -
1
fJ
+ fJ + 1jJ(-r) + 1jJ(-r)
b(-r) -
1
+ fJ
fJ
•
(51)
We are going to show that in the circle (C), we have
!d(-r)!
In fact we have
1 1 16·
<2
(52)
412
Appendtx A
+ 13 13
b(T) I
1
1
~ 21 b ('I') a
e"2
a
>2
=
2e
131
1..) 6 < 2(2 e% + e < e%' a
X 12 X 16,
- 131< 1.. ~ Ib(r) 1 + 13 4e 16'
(53)
From (51), (53) and (39), we see that (52) holds. By (46) we have (54) Hence from (43) and (47), we have
+
zoe'f'l (zoe') _ eN, 1 WI ('I') Nfl(zo) 1+13'
Then from (44) and (47), we see easily that if we write
(55)
we have
(56)
in the circle (C). n~l being an integer, we deduce from (45) and (46),
(57)
with c =
a.
(58)
Appendix A
413
By (46) and (29), we have
(59)
(60) and
Then making use of the inequality
log (1 -
x)
x >- I - x
(0
<
x
<
1)
we find
10 1
Ifl(zo)II<_2_<~<-.L.
g M(p,fl)
eO -
1
(61)
16
a
Study of the function f(z). From (18), (29) and (60), we have M
=
M(p,f)
=
AM(p,fl)
> A (M(p,f2) >aA >8.
-
I}
?
A (eO -
1)
(62)
From (60) and (62), we have
hence from (32), we have
(63)
Append~x
414
A
On the other hand, from (8), (50), (52), (55), (56), (57) and (61), we deduce
(64)
zoe'f' (zoe') -"""f(zo)
= N,{ 1 + .< e
Ul
(
)}
T
(65)
,
with
(66)
in the circle (C), and
(67)
'f(zo)'1 < 16' 1 1log M(p,f)
(68)
Let log (1 +Z) be the branch of the logarithm of 1 +Z, which is holomorphic for
Izl
and equal to zero at the point Z=O, Then we have
IlogO+Z)I~2IZlfor
1
IZI
Hence, by (66) and (68), the function
A(T) = log
If~o) 1 + log
(1
+ OCT)}
is holomorphic in the circle (C) and satisfies the inequality
Append~x
A
415
1
1).(-r) 1< 8'
(69)
Next, putting
we can write the formula (64) in the form
(70) in the circle (C), We are going to show that the circle (C):
1T 1
<
.ar::
contains the rectan-
16 V 3 gle
In fact, for
T
E (A), we have 1,,1 ~ IRe(,,) 1 + IIm(,,) 1 ~
4 N
+ 6" N'
and hence it is sufficient to verify that
16/3 (4
+ 6,,) < NO',
It is easy to see that this is true, by making use of (42) and (24), The function
s=
N"
+ ~I
maps the rectangle (A) onto the rectangle
416
AppendlX A
the interior and the boundary of (A) correspond respectively to the interior and boundary of (R' ), Moreover if /:" is a point on the boundary of (R' ) and /:, is a point of the rectangle
(R): -
2 < Re(t;)
1 <- 2'
0< [met;) <
47r,
we have
1
It;' - t;1 > 2' Consequently if /:, is a point of (R), then on the boundary of (A), we have
IN .. + II
-
t; I
1 > 2'
hence by (69), we can apply Rouche' s theorem and conclude that the function A( .. ) -
t;
=
(N ..
+ II
-
t;)
+ A( .. )
has a zero ,=,(1:,) in the interior of (A), The function ,=,(1:,) is holomorphic and univalent in (R) and A { .. (t;) } = t;
in (R), It follows that in (R) the function u (t;) = .. (t;)
+
(71)
lOO
is hoi om orphic and univalent with Re{u(t;)} = Re{ .. (t;)}
Moreover by (70) we have
<
0,
Append~x
417
A
(72)
in (R), Consider a number 0~w<2lt and the rectangle
(R~): -
2
<
Re(s)
1 <- 2'
w < [m(s)
< w + 27f,
We are going to show that the function eu(,J is univalent in (R~), In fact, assume on the contrary that there exist two points 1;.1,1;.2(1;.1#1;.2) of
(R~)
such that
Then by (71) and (72), we have e'(~')
=
e'(~2J,
T(Sl) -
T(S2)
e~' = e~2, Sl -
=
2pm
S2 = 2qm
(p ~nteger, p
# 0),
(q ~nteg er),
Next from (69) and (70), we have Sl
= N T(Sl)
+ ~I + ).{ T(Sl)} ,S2 = NIT(Sl) -T(S2)1
<
N T(S2)
IS1-S21
+ ~I + ).{ T(S2)} , 1 +"4'
On the other hand, since 1;.1 and 1;.2 belong to (R~), we have q
then by (32), we have
and hence Ip I <1. So we get a contradiction, Now consider (65) and define
= 0,
Appendtx A
418
which is holomorphic in the circle (C) and satisfies the inequality
(73)
The formula (65) can be written in the form
Then making use of the formula A(-r) = N-r
+ tl + A(-r),
we have
(74)
where
jJ.(-r) = - (1 = - (1 -
1
+ )"j(-r) -
1
+ ~(-r) + teNI
N»),,(-r) N)A(-r)
(1 -
1
N)tl -
+ t(l
-
()o)
()o)
is holomorphic in the circle (C) and satisfies the inequality
In (74) setting
,=,(0,
we get, in (R),
(75)
Appendix A
419
where
(76) is holomorphic in (R) and satisfies the inequality
1
Iv(S-)1<4+ 2n in (R). Theorem 2 is now completely proved. In what follows, we deduce some consequences of Theorem 2, in which the symbol a always denotes the absolute positive constant in Theorem 2.
Corollary 2. Let f(z) be !l holomorphic function in the circle Iz I<1 and S a positive number. Assume that there exists a number O
and that for each number O~w<2n, the domain
OJ
< arg w < OJ +
2n
is a simply covered image domain of f(z) for the circle Iz I <1. (see Definition 2. 1)
Proof. By Theorem 2, we can find p,N, u(e;) ,v (1:,) having the properties in Theorem 2. The function peu(t) maps the domain (Rw) onto a domain d w belonging to the circle
Iz I<po
Then by the formula
420
evidently fez) maps d", onto the domain (r ",).
Definition 4. Let F (z) be a function which is holomorphic and does not take the value zero in the circle
1
z 1 < 1. We denote by Log F (z) the branch of
the logarithm of F (z), which is holomorphic in the circle 1z 1 <1 and is such that O~Im
(LogF(O) }<2rr.
Corollary 3. Let F (z) be a function which is holomorphic and does not take the value zero in the circle
1
z 1 < 1. Let R be a positive number. Assume
that there exists a number O
Then we can find a number B
>R
+
ILogF(O)!>e-,:"o'
(77)
and a holomorphic and univalent function
(w) in the circle Iw-wol
1
1
<
1, F (
=
eW
in the circle Iw-wol
Proof. Putting f(z)=LogF(z) and S=4eR, then (77) implies M(To,f) ~ (4e)(S
+
If(O) 1 )e-,:"o'
Hence, by Theorem 2, we can find two numbers p, M such that
and a holomorphic function u (1:,) in the rectangle
-
2 < Re(t;) < -
1 2'
0 < [met;) < 4%
(78)
Append~x
A
421
having the property lOin Theorem 2. The function w =Me' is hoi om orphic and univalent in the rectangle
(Ro): -
2 < Re(t;) < -
1 2'
0 < lm(~) < 2"
and maps (Ro) onto the domain
Iw I <
(r o): Me- 2 <
1
Me-"2, 0 < arg w < 2".
The function w =Me' has then an inverse function 1;=L (w) which is holomorphic and univalent in (r 0). The function
¥..w) =
pe'(~), ~
= L (w )
is then holomorphic and univalent in (r 0) and we have
I¥..W ) I <
1,
f (¥..w )} = w, F (¥..w )} =
e'"
in (r o). Evidently (r o) contains the circle
and we have
Corollary 3 is therefore proved.
Corollary 4. Let F (z) be a function which is holomorphic and does not take the value zero in the circle
Iz I <
1. Let R be a positive number. Assume
that there exists a number O
422 + logM(ro,F)~27r(4e)2(1+R+log IF(O)I)e-I:,...
(79)
Then the conclusion of Corollary 3 holds.
Proof.
If IF(O) 1~I, then (79) implies (77), hence the conclusion of
Corollary 3 holds. If IF (0) I
(80) Let zo=roe tO , be a point of the circle Iz I =ro, such that IF(zo) I = M(ro,F)
> 1.
Consider the function
We have
where
Then by (80), we have M(ro,LogF 1) ~logM(ro,Fl) ~log IF1(~O) I
> (4e)2(R +
ILogF1(0) De-I:, •••
Append~x
423
A
Consequently by Corollary 3, we can find a number B>R and a holomorphic and univalent function <:pj(w) in the circle /w-wo/
in the circle /w-wo/
is holomorphic and univalent in the circle /w-wo/
/
<1, F{
W
in the circle /w-wo/
Corollary 5. Let F (z) be a function which is holomorphic and does not take the value zero in the circle / z / <1. Let K > 1 be a number. Assume that there exists a number O
+ logK + log+
/F(O) /)e-,:g,o'
(81)
Then for each number O~w<2lt, the domain
1
K
<
/Z / <
K,
(j)
< argZ <
(j)
+ 2n
is a simply covered image domain of F(z) for the circle /z/<1.
Proof. Putting R=logK+4lt, then (81) implies (79), hence the conclusion of Corollary 3 holds. We are going to show that the circle Iw-wol
with
log K < Re(w) < log K, 2ln < 1m (w) < 2ln +
4n,
424
AppendlX A
l
=
[:,J
In fact, if w is a point of (T), then Iw -
Bli ~ IRe(w)1
+
I/m(w) -
BI
+ 4,,= R.
Consider a number 0~w<2n and the rectangle (T ,,): - log K < Re(w) < log K , w
+ 2l" <
1m (w) < (w
+ 2l,,) + 2"
which belongs to (T). The function qJ(w) maps (Tw) onto a domain Dwbelonging to the circle Iz I<1. By the formula (78), it is easy to see that the function F (z) is univalent in Dwand maps Dwonto (C w)' Similarly we can prove the following corollary:
Corollary 6. Let F (z) be a function which is holomorphic and does not take the value zero in the circle I z I < 1. Let R be a positive number. Assume that there exists a number O
Then the conclusion of Corollary 3 holds.
Proof. If IF(O) I~l, the inequality in Corollary 6 implies (77), hence the conclusion of Corollary 3 holds. If IF (0) I> 1, then the inequality in Corollary 6 becomes
Let zo=roe iO , be a point of the circle Iz I =ro, such that
425
Appendix A
As in the proof of Corollary 4, consider the function
We have
Then we have
1
1
M(ro,LogF j ) ~ log M(ro, F j ) ~ log 'I-=F-j7(S-:-07)'1 1 logM(ro'/i)
=
+ R)e-,:'r. + ILogFj(O) I)e-,:.r•.
~ 2n-(4e)2(l
>
(4e)2(R
Then as in the proof of Corollary 4, we see that the conclusion of Corollary 3 holds. From Corollary 6 we deduce, as in the proof of Corollary 5, the following corollary:
Corollary 7. Let F (z) be a function which is holomorphic and does not take the value zero in the circle
Iz I <
1. Let K
there exists a number O
> 1 be
a number. Assume that
426
Appendix A
Then the conclusion of Corollary 5 holds. Consider a function F(z) which is holomorphic and does not take the value zero in the circle circle
Iz I<
Iz I <
1. Putting F = e', where f = LogF is hoi om orphic in the
1, the successive derivatives of F can be expressed as F'
F"
= =
eff' , ef(f'
FWI = ef(f'
2 3
+ f"), + 3f' f" + fW/),
and in general
(82) where P, is a polynomial of f' , f", .•• , f(') of degree
v -1
whose coefficients are
positive integers. Grouping together the similar terms of P .. we may assume that no two terms of p, are similar. Denote by Iv the largest coefficient of p, and by J, the number of the terms of p .. and define q(z) = P,(f' ,f","·,f('».
(83)
Corollary 8. Let F (z) be a function which is holomorphic and does not take the value zero in the circle Iz I<1. Let v be a positive integer and R a positive number. Assume that there exists a number O
M(TO ,Log F) ;;::0 c,(l
+ R + ILog F (0) I )e-,:.,,,
(84)
where c, is a positive constant depending only on v. Then we can find a number B >R and a holomorphic and univalent function qJ(w) in the circle Iw-wol
(85) in the circle Iw-wol
427 Proof. For the sake of definiteness we assume
v~2. The case of v= 1 is
much simpler. Consider the function f(z)=LogF(z) and a positive number S. Assume that there exists a number O
M(ro,f)~4e(S+ If(O)I)e-I:.".
(86)
Then by Theorem 2, we can find p,N ,u(O,v(O having the properties in Theorem 2. If (87)
S>e, we have 1~N<2(logM)2
and by the property 3 0 in Theorem 2, we have for l~n~v,
(88) T hen from (83) we see that in the circle
Iz I~p,
we have
(89) On the other hand, by the property 2 0 in Theorem 2, we see that in the rectangle (R) we have
(90) Consequently in (R) we have
(91)
Evidently we can find a positive number K. depending only on v, such that when M>K" the right member of (91) is less than 1/2. Hence if
428
Appendtx A
(92)
S>K" then in (R) we have, by (82),
(93) where Il (I;,) is a holomorphic function in (R) with
1
(94)
l.u(s) I <2 in (R), From (93) and the property 2 0 in Theorem 2, we have
(95)
in (R), where
g (s)
=
N v{log (pM)
1 fj)S
+ (1 -
+ v(s)} + Log (I + .u(s)}
is a holomorphic function in (R) and we have in (R), Ig (s)
I<
3vlog M
+ fJv,
(96)
where 13>0 is an absolute constant, Now consider the function Z=Me' which is holomorphic and univalent in the rectangle
(Ro): -
2
<
Re(s)
1 <- 2'
0
<
lm(s)
<
2"
and maps (Ro) onto the domain
(To): Me- 2 <
IZI <Me-t, O<argZ <2",
429
Appendtx A
Let ~=L(Z) b the inverse function of the function Z=Me'; L(Z) is holomorphic and univalent in (r o)' In (95) put
~=L
(Z), we get
(97) in (r 0), where 1/I(Z) =
pe'(L(Z))
(98)
is a holomorphic and univalent function in (r o) with 11/I(Z)
I <1
(99)
in (r o). On the other hand, HZ)
=
g {L(Z)}
(100)
is a holomorphic function in (r 0) with
Ih (Z) I <
3vlog M
+ fJv
(101)
in (r 0)' Evidently (r 0) contains the circle
IZ-Btl<'Q, Consider a point w of the circle T : Iw - Bi I<~ and a point Z on the circle IZ- Bi
I =Q.
Then Q
IZ - wi> 2' On the other hand, when M>K~, where K~ is a positive number depending only on v, we have
430
Appendix A
3vlogM
+ pv < %.
Hence if S>K~,
then for each point wE r
,
we have Iz - w I
Ih(Z) I <
on the circle IZ- Bi I =Q. Consequently by Rouche' s theorem, the function Z+ h (Z) -w has just one zero Z(w) in the circle IZ-Bi I
r.
In (97) putting Z=Z(w), we
get
in
r , where the function CP(w) = 1/1 {Z (w)}
is holomorphic and univalent in
r
with
lCP(w) I in
<1
r. Finally let m,=max(e,Kv,K~) and taking
S = m,
+ 16eR,
(102)
then Q
B
>"2 >
M
16e
>
S
16e
>
R.
Append~x
431
A
Consequently if we take in (84), c, = 4e(m,
+ 16e),
then (84) implies (86) with S defined by (02), and hence the conclusion of Corollary 8 holds. In what follows, cvalways denotes the positive constant in (84).
Corollary 9. Let F (z) be a function which is holomorphic and does not take the value zero in the circle jzj
logM(To,F) ~ 0
+ 2n-)vc,O + R + log+
jF(O)j
+ log
1
a
-1--)e-togro' -
TO
(03) Then the conclusion of Corollary 8 holds.
Proof. If jF(O)
j~l, then
(03) implies (84), hence the conclusion of
Corollary 8 holds. If jF(O) j <1, then (03) becomes
[ogM(To,F) ~ 0
+ 2n-)vc,O + R + log
zo=roe"o being a point of the circle jz j =ro, such that jF(zo) j = M(TooF)
Consider the function
> 1,
1
a
-1--)e-to• r o' -
TO
•
(04)
432
Append,x A
We have
1
where
Zo -
So = 1 _
Z] Iz]1
To - T].. T] e' 0,
= 1_
I I So
< To·
Then by (04), we have M(TO,LogF l ) ;;?; log M(TO,F l ) ;;?; log IFl (so) I > logM(To,F)
> c,O + R +
ILogFl(O) I )e-I:.,o.
Hence by Corollary 8, we can find a number B>R and a holomorphic and univalent function q>l(W) in the circle Iw-wol
in the circle Iw-wol(W)=Zl+q>l(W)O-lzll) is holomorphic and univalent in the circle Iw-wol
in the circle Iw-wol
Corollary 10. Let F (z) be a function which is holomorphic and does not take the value zero in the circle IZ I <1. Let v be a positive integer and K > 1 a number. Assume that there exists a number O
(01)
Append~x
A
433
where c~=4Jl(l+2Jl)vcv. Then for each number O~UJ<2Jt, the domain
(Co» :
1 K < 1z 1 <
is a simply covered image domain of
K, UJ < aTg Z < UJ + 2n
F(v)
(z) for the circle
z 1 <1.
1
This corollary is deduced from Corollary 9 by the method used in the proof of Corollary 5. Similarly we can prove the following corollary:
Corollary 11. Let F (z) be a function which is holomorphic and does not take the value zero in the circle
1
z 1 <1. Let v be a positive integer and R a posi-
tive number. Assume that there exists a number O
+ 2n)vc.(l + R
+ 1 + log IF(O)
1
1 a + log 1 _ TO)e- 1. . .,.
Then the conclusion of Corollary 8 holds.
Proof. If IF(O)
1~1, then the inequality in Corollary 11 implies (84),
hence the conclusion of Corollary 8 holds. If IF(O)
1>1,
then the inequality in
Corollary 11 becomes
1
1
a
10gM(To'-F) ~ (1 + 2n)vc.(l + R + log -1--)e- 1. , . , . - TO
Let zo=roe i9 , be a point of the circle
1z 1 =ro,
such that
As in the proof of Corollary 9, consider the function
Append~x
434
A
We have
Then we have 1 1 M(1'o,LogFl)~logM(1'o'Fl)~log IF1(S-0)1 =vlog(l-Izll> = vlog(l- IZll> > vlog (1 -
1'0)
+ log
1
1
IF(zo)1 >vlog(l-1'o) +logM(1'o'Ji)
+ c,(l + 2n- + R + vlog + c,(2n-v) (log
> c,(1
+ 2n- + R + vlog
1 ,o"gr• -1--)e- 1'0
1 " -1--)e-,ogr. - 1'0
1 " -1--)e-,ogr. -1'0
Then we complete the proof of Corollary 11 as in the proof of Corollary 9. As for Corollary 10, from Corollary 11 we can deduce the following corollary:
Corollary 12. Let F (z) be a function which is holomorphic and does not take the value zero in the circle
Iz I <
1. Let
v
be a positive integer and K
number. Assume that there exists a number O
where c~ = 4n (1
+ 2n)vc,.
+ log K + log+
1 IF (0) I
+ log
1
_"_
1 _ 1'0)e- 10gr.,
Then the conclusion of Corollary 10 holds.
>1 a
435
Appendtx A
3. GENERALIZATION OF THEOREM 2 Theorem 2 can be further generalized as follows:
Theorem 3. Let Hz) be a holomorphic function in the circle Iz I <1, k a positive integer and S a positive number. Assume that there exists a number 0< ro< 1 such that
Under these conditions, we can find two numbers p, N such that
_ e '.
and k
1, M = M (p ,f)
+ 1 holomorphic functions (R): -
> S,
k ~ N
< 1 + (log
M S)2
u (/;), vp (I;,) (p = 1 ,2, ••• ,k) in the rectangle
2 < Re(t;) < -
1 2'
0 < 1m (~) < 2(k
+ Un
having the following properties: lOIn (R), u (/;) is univalent and Re {u U;)} <0. For each number 0~w<2rr, e
U
(')
is univalent in the rectangle
(R",): -
2 < Re(~) < -
1 2'
w < Im(~) < w
In (R) we have
f
{pe'(~)}
=
Me~.
2 0 For l~p~k, in (R) we have
Iv,(~)
1
1< 2n- + 2
+ 2kn".
436
Appendix A
and
3 0 For n~l, we have
In this theorem, ak ,b k ,Ck are positive constants depending only on k. For the proof of Theorem 3, which is quite similar to that of Theorem 2, see [1 OJ, where Theorem 3 in a more complete form is proved. In what follows, we deduce some consequences of Theorem 3, in which the symbol ak always denotes the positive constant in Theorem 3.
Corollary 13. Let f(z) be a holomorphic function in the circle Izl
M(ro,f)~4e(S+
H
E
• -0
If(')(O)
n!
1
a.
)e-1og,o .
Then we can find a number M such that
and that for each number 0:::;;;w<21l, there are k disjoint simply connected domains Dj(w) (j=1,2,···,k) belonging to the circle Izl
Me- 2 < Iw
1
1
<Me-z,
w<argw <w+ 2n.
Proof. By Theorem 3, we can find p and u (1:,) having the properties in
Appendtx A
437
that theorem. We are going to show that the number
M = M(p,f) >S satisfies the required condition. In fact, consider a number O:::;;w
-
pe
U
(,)
Z < Re((;) < -
1 2'
peu(,)
w
is univalent in the domain
+ Z() -
1)n< ImU;) < w
+ Z)1f,
maps this domain onto a simply connected domain D j (w) belonging to the
circle Iz I <1. By the formula
we see easily that Hz) is univalent in D, (w) and maps Dj (w) onto (r oJ. Since pe u (,) is univalent in (R w ) , evidently D j, (w) and Dj, (w) have no common point, when jJ7i:j2.
Corollary 14. Let Hz) be a holomorphic function in the circle
Iz I <
1.
Let k ,p ,d be positive integers and R a positive number. Let F(z) be a function of the following form:
F(z)
=
ef('){Qo(z)
+
,L'Q,(z)},
(105)
i=l
where p
Qo(z) = aoII (j(j»)m"
Q,(z) = a,II (j())m'i
(t = l,Z,···,n)
(06)
in which ao#O, a,Ci=l,Z,···,v) are constants and m;Cj=O,l,···,k), m'jCi=l,Z, ..• ,v,J=O, 1 , •.. ,p) are non-negative integers with I
17 m) j=O
=
P
d,
17 m 'J j=O
(t
=
1, Z , ••• , v) •
(107)
438
Appendix A
Assume that there exists a number O
(lOB)
where a is a positive constant depending only on k,p,d and
(109)
Then we can find a number B>R and a holomorphic and univalent function
qJ
(w) in the circle Iw-wol
I
(10)
in the circle Iw-wol
Proof. The proof of this corollary is quite similar to that of Corollary B. The main difference is that now we base upon Theorem 3 instead of Theorem 2. Following the procedure of the proof of Corollary B, consider the function Hz) and a positive number S. Assume that there exists a number O
Then by Theorem 3, we can find p,N,uC1;,),v p (t,)(p=1,2,···,k) having the properties in Theorem 3. If
S>e,
011)
we have k:S;;; IV
<
2(logM)2
and by the property 3 0 in Theorem 3, we have for O:S;;;n:S;;;p,
(112)
439
Appendtx A
T hen from (106) and (107), we see that in the circle IQi(Z)
I ~ la, IB.",dMd-1 (log M)2 p(d-IJ,
Iz I~p,
we have
B',P,d = A~:;l
(13)
(t=1,2,···,v). Hence if we put
, Q(z) =
1:Qi(Z), i=l
then in the circle
Iz I~p,
we have
,
IQ (z) I ~ B"p,dMd-1 (log M)2 p(d-IJ 1: Iai I,
(14)
t=1
On the other hand, from the properties 10 and 2 0 in Theorem 3, in (R) we have for O~j~k,
Hence in (R) we have
(15)
From (14) and (15), in (R) we have Q{pe 'W} I B ' < '",d IQo{pe'(O}
Let
f'k ,p,d
I_a'i
(logM)2,(d-IJ., 1: M ,~I ao'
B' = B e(2+2K+~Jd ',p,d '",d
be a positive number such that when M >Ak,p,d' we have
B''.P .d
(log M)2,(d-IJ I
M2
1
<-2'
440
Appendtx A
Then we see that if
S
>
A,.,.d' S
ai I )2 > (i~l, I ~
,
(16)
we have in (R),
(17)
By the properties 1° and 2° in Theorem 3, we see that for O~j~k, we have in (R) ,
(18)
where V j (/;) is a holomorphic function in (R) such that
Iv/s-) I < in (R),
r.
I,
+ klogN,
(19)
being a positive constant depending only on k. Consequently from
(18), (19) and (06), we have in (R),
(20) where V (I;,) is a holomorphic function in (R) such that
IV(S> 1< d (I, + klogN)
(21)
in (R). From (05),017),020) and the property lOin Theorem 3, in (R) we have
(22) where I (/;) is a holomorphic function in (R) such that
441
AppendIx A
11(S-)
I<
(123)
1
in (R). (122) can be written in the form
(124) where 9 (S-)
=
Logao
+ dlog M + V (S-) + 1(0
is a holomorphic function in (R) and by (121),(123) and (112), we have
Ig
(0 1< d (1 r;.d =
+ 2k)/ogM + I/og laoll + r;.d' 27T + 1 + d (J, + klog 2)
(125)
in (R). Comparing (124) with (95), evidently we can continue the proof in the same way as in the proof of Corollary 8. We see that if
S
> A;.d, S > e-z 16 - eI
2
I/og Ia 0 II
,S>
16eR ,
(126)
then we get the conclusion of Corollary 14. Hence in (108), it is sufficient to take a
=
4eA"p,d'
where A',p,d
=
max (A"p,d
,X", ,32e).
APPENDIX B SOME THEOREMS ON MEROMORPHIC FUNCTIONS
1. A THEOREM
ON MEROM ORPHIC FUNCTIONS IN THE
PLANE For the sake of convenience, we first give the following definition: (see Definition 3.3) Definition. Let f(z) be a merom orphic function in C and a E C a value. Let m~2
l-~ m
be an integer or m =
+=.
We say that a is an exceptional value of weight
of f(z) in C, if in C the equation fez) =a has no root whose order of multi-
plicity is less than m. In particular, if m =
+=, an exceptional value a of weight
1 of f(z) in C is such that the equation f(z) =a has no root in D.
Theorem 1. Let ajEC(j=1,2,···,q;3~q~5) be q distinct values and mj ~2 (j
= 1, 2,,,, ,q) be such that for each 1 ~j~q, mj is an integer or
+=,
and
that
(1)
Then there does not exist a non-constant meromorphic function f(z) in C such that for each
l~j~q,
the value aj is an exceptional value of weight
l-~ mj
of f(z)
in C.
Proof. Assume on the contrary that there exists a non-constant merom orphic function f(z) in C such that for each l~j~q, the value aj is an exceptional
443
Appendtx B
444
value of weight 1--.l of Hz) in C. By the second fundamental theorem in Nevanm;
linna's theory of meromorphic functions [13J, we have the inequality q
+ S(r,j).
(q-2)T (r ,j) ::::;; ~N(r,a)
(2)
j~l
In general
N (r ,a) is defined by the formula
-(
N r,a)
where
=
1r n(t,a)-n(o,a) dt + n-( o,a)logr, t
o
n(t ,a) denotes the number of the roots in the circle
(3)
Iz I::::;;t of the equation
f (z) = a, each root being counted once. On the other hand, S (r, f) denotes a function of r, such that there is a set a of values of r of finite linear measure, such that
ltm S(r,j) = O. T (r,j)
r-H=
(4)
r'"
By the assumption, evidently for each l::::;;j::::;;q, we have
1 N(r,a)::::;; -N(r,aj)' mj
(5)
In general N (r ,a) is defined by the formula
l
r
N (r,a) =
n(t,a)-n(o,a) t dt
+ n(o,a)logr,
(6)
where n (t ,a) denotes the number of the roots in the circle Iz I::::;;t of the equation
Hz) =a, with due count of order of multiplicity. By the first fundamental theorem in Nevanlinna' s theory of meromorphic functions, we have
N(r,a)::::;; T(r,j) From (2), (5), (7), we deduce
+ 00).
(7)
Appendtx B
445
(i:o- ~)-2}T(r,f):::;; 00) + S(r,f). j=l
}
This inequality is impossible, in view of (4) and (1). So we get a contradiction.
2. A THEOREM ON MEROMORPHIC FUNCTIONS IN THE UNIT CIRCLE Theorem 2. Let k?l be an integer. Then we can find a positive constant A. depending only on k having the following property:If Hz) is a meromorphic function in the circle Iz I <1, such that each of the equations fez)
= 0,
f(')(z)
= 1
(8)
has no root in the circle Iz I <1, then in the circle Iz I < 312 one of the inequalities
holds.
1
This theorem was first proved by Gu[17](for the circle Izl<64). Its proof was later simplified by Yang[38]. In what follows we give a complete proof of Theorem 2. For this we need some lemmas.
Lemma 1. Given an integer k?l, we can find positive constants A;Cj=l, 2, ••• ,7) having the following property: If f(z) is a meromorphic function in a domain Izl
(9) than for O
+
+ 1 A 2 log -
~r
+
+ + 1 A3log p+ A4 lo g r
Appendix B
446 +
+
+ A5 10 g log
1
-r;;:T
+
+ A6 10 g + A 8
(0)
7•
Proof. For k = 1, Lemma 1 is known [13J. We are going to prove it by mathematical induction. Assume that it is true for k = 1,2,"',p, and let us show that it holds for k =p
+ 1.
In fact, let Hz) be a meromorphic function in a do-
main Izl
We have
where a,(j=1,2,"',p) are positive integers. Hence
and we have 9 (,+1)
f(,+lJ
m(r'-f-)!'( m(r'-g-)
p
p
where ~ = L J and ~ I = L log I a, I j=l
j=l
+
,
f<})
~m(r'T)
+ log (p + 1).
+
+ Jllog
1 -;:
+ Jl'
,
(12)
Since by assumption Lemma 1 is
true for k = 1,2,"',p, it follows from (12) that it is sufficient to show that for 0
m(r,g--) 9
<
+
+
B1log T (p,g) +
+
+ B5 lo g log
1
-r;;:T
+ B2log
-
1
+
+ B6 lo g
+
~
(8
+
+ B310g p + B4log
+ p) + B
1 r
7,
where Bj(j= 1,2,,,,,7) are positive constants, in view of the inequalities
(13)
Append~x
+
B .
447
+
T (p,g ) ::( p log p + T (p,!) ,
+
log (s + p) ::( log s + log (2p).
In fact by the identity 9 (,+1) -g-
9 (,+1) 9 (,)
= g -g-
= (r+p) /2,
we have for O
9 (,+1)
9 (,+1)
m (r , - - ) ::( m (r ,-(-,)-) 9 9 (,+1)
9
+
m (r ,-(-,)-) < a] log T (p' ,g (,» 9
+
+
+aslog log
9 (,)
+ m (r , -9 ) ,
(14)
+ 1 + + 1 + a2log - , - + a3log p' + a,log P
1
IAc.1
-T
T
+
(15)
+aslog s+a7t
where A= (s +p) (s +p-1) ... (s + 1) and aj (j = 1,2, ... ,7) are positive numerical constants, and by assumption, 9 (,)
+
+
1
+
+
1
m(r,-)
where B; (j= 1, 2, ···,7) are positive constants. Consider (15). We have
+ (p + 1)N(P' ,g) + 1)T(p,g) + m(p' ,g-), (,)
::( (p +
+
+
9 (,)
log T (p' ,g (,» ::( log T (p,g) + log m (p' , L ) + log 2(p + 1), 9 +
(,J
9 (,)
+
log m(p' , L ) ::(m(p' , - )
9
(16)
Appendix B
448
where H denotes the right member of (16). Hence + log T(P' ,g('»
+
+ H + p,
(17)
where f3 is a positive constant. (14), (15), (16), (17) yield (13). Evidently in Lemma 1 the positive constants Aj (j = 1,2, "',7) depend only on k.
Lemma 2. Let k;? 1 be an integer and Hz) a meromorphic function in a domain Izl
00.
Then for O
have f(O) + + 1 + + 1 m(T'-f )
+
1
+
+ Aslog log If(O)
1
T
+ A 6,
(18)
where A j (j=1,2,···,6) are positive constants depending only on k. This is a particular case of Lemma 1.
Lemma 3. Let k;? 1 be an integer and Hz) a meromorphic function in a domain Izl
Then for O ) < B} log T (p,f) + B 2 /0g + B3/0g P + B4 lo g ~T
+ + 1 + Bs/og log If(O)
1
+
+ + 1 B6 /0 g log If(O)(O)
T
1
+B
7 ,
where B j (j=1,2,"',7) are positive constants depending only on k.
Proof. First by Lemma 2, for O
(19)
449
Appendtx B
f
eHll
m (r 'J<'l)
< a] log+
T (fl ,f'l») +
+ a2log+
+
1
fI-r
+ a3 lo+g
+ 1 fI + a4log -:;:
1
(20)
+aslog log Ifel)(O)1 +ap
where aj (j = 1 ,2, ..• ,6) are positive numerical constants. Next as (17), we find + + log T(p' ,fell)
+ L + a,
(21)
where L denotes the right member of (18) and a is a positive constant depending only on k. From (20) and (21) we deduce (19). Similarly we prove the following lemma:
Lemma 4. Let k ~1 be an integer and f(z) a meromorphic function in a domain
Iz 1
such that
f(O)"# 0,00, fel)(O)"# 1.
Then for O
f eHll + + 1 + + 1 m(r'fe1) 1) < B]log T(p,f) + B2log + B3 lo g p + B4 lo g ~r r ++ 1 + Bslog log If(O) 1
where Bj(j = 1 ,2, .•• ,6) and
B~
++
+ Bslog log
1 Ifel)(O)-ll
+B
, 7 ,
are positive constants depending only on k.
Lemma 5. Let f(z) be a holomorphic function in a domain 1 z 1
Then for O
This lemma is known[13J.
Lemma 6. Let U (r) be a non-negative and non-decreasing function in an
450
Append~x
B
interval ro
U (r)
+
< a log
U (p)
+ alog
L
p-r
+ b.
(22)
Then for ro
U (r)
<
2alog L
p-r
+ 2b.
(23)
Proof. In the case ro= 0, this lemma is known[13]. Consider the case ro> O. Let V (r) be the non-negetive and non-decreaseing function defined for O
Lemma 7. Let U (r) be a non-nagative and non-decreasing function in an interval O
(24)
where A j (j=1.2,3,4) are positive constants. Let ro(O
U (r)
where
<
2Alog L
p-r
+ 2B,
(25)
Append~x
B
451
Proof. For ro
U (r)
+
<
A log U (p)
+ Alog Lp-r + B.
Hence by Lemma 6, for ro
Lemma 8. Let a>e and x>O be two numbers. Then we have
logx
+ a log+
+ 1
log - ~ a(loga-1) x
+ log+
x.
This lemma is known[13J.
Lemma 9. Let Hz) be a meromorphic function in a circle Izl
fez) = 0, f(1)(z) = 1
has no root in the circle
qi.z)
=
Iz I
f(1)(z), ?fez)
(26)
Put
=
(k
+ 2) qi.z)-l ({J' (z) -(k + 1) ({J"(z) ({J' (z)
(27)
and assume f(O)
Then for O
=1=
00,
({J'
(0) =1= 0, ?f(0) =1= O.
(28)
452
Appendix B
1 + (2 + T)log
If(O){(CP(O)-I} gJ' (0)
I+ Tlog 1
1 11/>(0) 1
+ a ..
(29)
where a. is a positive constant depending only on k. Concerning this lemma see [15].
Proof. We start from the following identity of Bureau:
We have
Since
1
T(T,f) = m(T,y)
= gJ- 1 m(T,--,-) gJ
=
1 m (T , y)
1 + N(T,y) + log
+ log
gJ' m(T'--I) gJ-
1
f ( 0)
If(O)
1
1,
gJ' gJ- 1 + N(T'--I)-N(T,--,-) gJgJ
+ log I~,O~O~ I ' and by a formula of Nevanlinna[28], gJ' gJ-l N(T'--I)-N(T,--,-) gJgJ
=
1
N(T,gJ' )-N(T,-,) gJ
1 + N(T'--I)-N(T,gJgJ=
hence
-
1
N(T,f)-N(T,-,), gJ
1)
Append~x
B
-
453
1
T (r ,f) ~ N (r ,f)-N (r , - , ) gJ
+ S(r),
(31)
where gJ
gJ' gJ' + m(r'j) + m(r'gJ_ 1) I+ l og 2 • + l og If(0){gJ(0)-1} gJ' (0)
S(r) = m(r,y)
(32)
It remains to estimate N (r, 0. First of all, if we denote by N 1l (r, 0 the
counting function for the simple poles of f(z) and by N(2(r ,f) the counting function for the multiple poles of Hz), each multiple pole being counted only once, then evidently (33)
Put
g (z) = gJ(z) -
1 = f(i)(Z) -
g' (z) + 2) - g (z)
1, 1/J(z) = (k
(k
+ 1 g')g "(z) -(z) (34)
and consider a simple pole Zo of Hz). Then in a circle
f=_c_+ p , z -
where p=p(z) is a holomorphic function in
m
T g
= (-
=f
= (_
ck! )HI (1
1)i ( Z -
Zo
1)i (Z -
+ Q),
Q
= (_
1)i+1 c(k (Z -
+zo)H2 1)! + p(i+1l =
"= (_
1)H2 c(k (z -
+ZO)i+3 2)! +
g'
g
(i)
c
Zo
p(H2)
r.
we have
=F 0,
Then in
ck! zO)HI
=
r : Iz-zo I
r
we have successively
+ p (i) ,
c' (Z -
ZO)i+1 (p (i) -
1),
(_
1)i+1 c(k (Z -
+Zo)H2 1)! (1 + Q ), 1
= (_
1)i+2 c(k (z -
+ZO)i+3 2)! (1 + Q ) 2 ,
Append~x
454
B
hence
+ 1 1 + Q\ = (_ 1) k + 1 (1 + Q\ - Q) 1+Q 1+Q k + 1 {l + (z - zo)'+\R\} = (- 1) k + 1 + (z -
L = (_ g
= (-
Consequently
1) k
Z -
1)
Z -
Zo
Zo
Z -
Zo
Zo
zo)'c;R\9
Zo
Z -
is a zero of 1\' (z), the order of multiplicity of
Zo
is at least equal
to k. It follows that
Let us estimate N(r,1\'), By (39), the poles of tV(z) are all simple poles which are the zeros and poles of g (z) and g' (z). But by hypothesis of Lemma 9, g (z) has no zero and the poles of g (z) and g' (z) are the poles of f(z). On the other hand, we have shown that the simple poles of Hz) are zeros of tV (z). It follows that
-
N (r ,1jJ) ~ N (2(r,f)
1 + -N (r,,) g
=
-
N (2(r,f)
+ -N (r , -1, ). ~
(36)
From (35) and (36) we get
1-
N\)(r,f) ~ T{N(2(r,f)
1 1 + -N(r'g;;-)} + T{m(r,1jJ) + log
1
11jJ(O) I}' (37)
On the other hand, by (31) we have
N\)(r,f)
+ 2N(2(r,f) ~N(r,f) ~T(r,f) ~N(r,f)
-
1 N(r,-/) ~
+ S(r)
455
Appendtx B
hence
-
N(2(r,f)
~-
1
N(r,-,) rp
+ S(r).
(38)
From (37) and (38) we have
(39)
and then
-
N(r,f)
=
NlJ(r,f)
1 1 + -N(2(r,f) ~- N(r,-;p!) + (1 + T)S(r) 1
+ T{m(r,1jJ) + log
1
11jJ(0)
I}·
(40)
(31) and (40) yield
1
T(r,f) ~- 2N(r,-;p!) ~ (2
1 1 + (2 + T)S(r) + T{m(r,1jJ) + log
1 1 + T)S(r) + T{m(r,1jJ) + log
1
11jJ(0)
1
11jJ(0)
I}
I} (41)
By (27) we have , m(r,1jJ) ~ m(r'~l) rp-
qJ'
+ m(r,-, ) + log2(k + 1)(k + 2). rp
(42)
Finally from (41),(32) and (42), we get (29). N ow let us come back to the proof of Theorem 2. For the sake of simplicity, in what follows we denote by 0.,0.' ,0.]90.2,0.39···,13,13' ,13]9132,133"·· and I ,I',
r
l'
1
2,
13 positive constants depending only on k, which, in particular, may be
numerical positive constants.
456
Appendix B
Consider a meromorphic function f(z) in the circle Izl
1 such that each of the equations (8) has no root in the circle Izl<1. Assume first that there exists a point
Zo
of the circle Iz I <1/8, such that
where q>(z) and w(z) are defined by (27). Consider the function F(z)
=
+ z)
j(zo
(44)
which is meromorphic in the circle I z I <7/8 and is such that each of the equations F(z)
= 0,
F(i)(Z)
= 1
(45)
has no root in the circle Iz I <7/8. Put
t[!(z)
=
F(i)(Z), 1J! (z)
=
(k
t[!1 (z) + 2) t[!(z) _ 1 -
(k
+ 1) t[!1t[!1I(Z) (z)'
(46)
Then from (43), we have
Hence by Lemma 9, we have for 0
T(r,F) ~ (2
1
t[!
1 t[!11 + Tm(r,~) +
(2
1
+ T)[og
2
t[!1
+ T){m(r'F) + m(r'F)} + /F(O){t[!(O)-l}/ t[!1 (0)
(2
+ T)m(r't[! 1
+ Tlog
1 11J!(0)
t[!1 _ 1)
1+ ai' (48)
Applying Lemmas 2,3,4 to F (z), we see, by (47), that for 0
T(r,F)
< P{log+
T(p,F)
+ log
1
--
p -
r
+ log
1
-
r
+
1
Appendix B
+ + 1 + log log IF(O) I
+
+ log
457
+ 1 log 14'>(0) -
+ (2 + t)log IF(O){4'>(O)
-
II}
I + p'.
1}
(49)
We have 1 1 T(r,F) +log IF(O)I = T(r'[i)
(50)
and by (47), + + log T(p,F) ::;;'Iog (T(p,F)
+ log
al + 1 IF(O)I} ::;;'Iog T(p'[i)
+ log+ + log
al
+ log2. (51)
On the other hand, by Lemma 8 and (47), we have
+
1
+
1
O+T)/ogIF(O)I+Ploglog IF(O)I::;;'Pp
(2
1 + T)log 14'>(0) -
11
+ P log+
From (49)-(53), we deduce that for
1 T (r '-F)
<
+ 1 p{log T (P'-F)
+ 1 log 14'>(0) _ 11::;;' P2·
0
F
<
1 + log - + log p- r
alog - p p-r
In particular for r=1/2, p=3/4, we get
+ at .
(53)
we have
1 -}
r
+ p".
Applying Lemma 7 to T(r,t) with ro=t, we have for ro
T (r ,1.)
(52)
(54)
458
Next applying Lemma 5 to l/F(z), we get +
log M(
Now consider a point
Zl
1 1
4" 'F) <
IZ1 I <1/8.
such that
3a".
Since
we have
Consequently in the circle
Izl<1/8,
we have
1 I< e Ifez)
30"
(55)
.
In order to complete the proof of Theorem 2, we distinguish several cases. A. In the circle
Iz I < 1/8
we have .+1
~
If(J)(z)
I~
j~O
t-
(56)
Distinguish again two cases: A 1. In the circle
Izl<1/32,
A 2 • In the circle
Iz I<1/32
we have always
there is a point If(zo)
As above consider the function
I=
1.
IfCz) 1::::;;1
Zo such
or always
IfCz)
I~
that (57)
459
Appendix B
F (z)
which is meromorphic in the circle
= f
(z 0
+ z)
Iz I<31/32.
By (56), in the circle
we have
and hence
Consequently for O
and then by Lemma 2 and (57), for O
+
{log T (p,F)
+ 1 = I (log T (P'-F)
+ log + log
-
1
p-r -
1
p-r
+ log + log
Applying Lemma 7 with
ro =
1 64'
5 r = 64'
we get
T (654
and then by Lemma 5, we have
,j;-) < If
11 p = 128'
1
+ l}
-
r 1
-
r
+ I}.
Iz I<3/32
460
Appendtx B
+ 1 1 log M(16'F) ( 1 1) M 16'F
which implies that in the circle
<
91' ,
< e91',
Izl
(58)
91'
B. In the circle
Iz I <1/8 there is a point .+1
~
If(J)(zo)
Zo
I<
j-O
such that
i-.
(59)
Distinguish again two cases: B I • There is an integer j(1~j~k +2) such that in the circle
Iz I <1/8 we
have
(60) Then for
Izl
+ r 'f(j)(t;)dt;, J"
i- + i- t, 1< i- + t i- < t,
If(j-l)(z) If<J- 2 )(z)
I<
=
1
If(z)1 <2·
B 2 • There does not exist an integer j(1~j~k+2) satisfying the condition in case B I • Then there is a point
Zl
of the circle
Iz I <1/8, such that
461
Appendix B
(61)
and that on the segment ZOZJ we have
(62)
As in the above, we find successively that on ZOZJ we have
If(J)(z)l
(63)
()=O,I,···,k).
Distinguish again two cases: B~.
If(k+2)(zJ)
1?1/2.
In this case we have
and
I (k
+ 1)f(H2)(ZJ) (f(')(zJ)-l}-(k + 2){f('+J)(zJ) }21 :>-k+l_k+2_3k+2
:::--
4
16
-
16
.
Hence the condition (43) is satisfied and consequently in the circle IZ I <1/8 we have an inequality of the form (55), namely
1 1< e3,·. 1f (z) B;.
If(k+2) (zJ)
1< 1 /2.
(64)
T hen there is a point Z2 of the circle IZ I
< 1/8,
such
that
1 2
(65)
Appendix B
462 and that on the segment
ZlZ2
we have
(66)
From (61) and (66), on
ZlZ2
we have
l4 _ l l< 2 4
If(Hl)(Z)
1< l4 + II 2 4'
(67)
On the other hand, by (59) and (62), we have
and then by (67), on
ZlZ2
we have
(68)
Next from (63) and (68), we find successively that on If(j)(z)1
<
ZlZ2
we have
1 (j = O,l,"',k-1),
(69)
Hence we have
If(Z2)
I<
I(k
1, If(1)(Z2)
I< ~
123,
+t <
If(H1)(Z2)
I<
+i-
+ 1)f(1+2)(Z2){f(1)(Z2)-I}-(k + 2){f(Hl)(Z2»)21
1 1 13 > (k + 1) 2(1 - 42 2) -
(k
+ 2) 421 49
=
1
43 00k
+ 1),
The condition (43) is therefore satisfied and consequently in the circle
IZ 1< 1/8,
we have an inequality of the form (55), namely
(70)
Appendix B
Finally from (58), (64) and (70), if we put
a. = max (9T' ,3T" ,3T"'), then the positive constant A. =e'\ has the required property in Theorem 2.
463
REFERENCES 1. W. Blaschke, "Eine Erweiterung des Satzes von Vitali tiber Folgen anaIytischer Funktionen", Leipzig Sitz. ber. , 87(1915). 2. A. Bloch, "Les theoremes de Valiron sur les fonctions entieres et la theorie de I' uniformisation", Annales Fac. sc. Toulouse 17(1925) 3. A. Bloch,
"Quelques
theoremes
sur
les
fonctions
entieres
et
meromorphes d ' une variable" , Comp tes rendus 181 (1925) Comp tes rendus 182 (1926). 4. H. Bohr, "Uber einen Satz von Edmund Landau", Scripta Univ. Hierosolymitanarum 1 (1923).
5. C. Caratheodory, "Sur quelques generalisations du theoreme de M. Picard", Comptes rendus 141 (1905). 6. C. Caratheodory, "Sur quelques applications du theoreme de Landau-Picard" , Comp tes rend us 144 (1905). 7. C. Caratheodory, "Beitrage zur Konvergenz von Funktionen Folgen", Sitz ber. Berliner Akad. (1991).
8. C. T. Chuang," A generalization of a theorem of Montel", Science Reports of the National Tstnghua Universtty 3 (1935) 215-220.
9. C. T. Chuang, "Etude sur les families normales et les families quasi-normales de fonctions meromorphes" , Rend tconti del Circolo Matematico d i Palermo 82 (1938).
10. C. T. Chuang, "Un theoreme general sur les fonctions holomorphes dans Ie cercle unite et ses applications", (I) Scientia Sinica 6 (1957) 569-621 11. C. T. Chuang, "Generalizations of the notion of normal families of meromorphic functions", Contemporary Mathematics, 25 (1983) 1-18. 12. C. T. Chuang, "On the theory and applications of Qm-normal families of meromorphic functions", Complex Analysts I, Proceedtngs, University of Maryland (1985-86)44-77. 13. C. T. Chuang and C. C. Yang, Fix-Points and Factorization of Meromorp lite Functions, World Scientific Publishing Co. , Singapore (1990).
465
466
References
14. P. L. Duren, Univalent Functions, Springer-Verlag (1935). 15. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford (1964). 16. G. Julia, Lecons sur les Fonctwns Uniformes
a Point
S~nguher Essent~el
I sole, Imprimerie Gauthier-Villar, Paris (1924).
17. Y. X. Gu, "A criterion of normlity of families of meromorphic functions",
Sc~ent~a S~ntca
18. Y.X.Gu, Normal
Spectal Issue (1) (1979) 267-274.
Fam~liesof Meromorph~c
Functions (in Chinese), Ed-
ucational Press of Szechuan (1991).
19. E. Landau, "V eber eine Verallgemeinerung des Picardschen Satzes" , S~tz.
Kg.Akad. Wiss., Berlin (1904).
20. E. Landau, Darstellung und Begrundung
e~n~ger
neuerer
Ergebn~sse
der
Funktwnentheone (1929).
21. A. J. Macintyre, "Wiman's method and the' flat regions' of integral functions", Quart. J. of Math. , 9 (1938).
22. A. J. Macintyre, "On Bloch's theorem", Math. Zeit. 44 (1939). 23. F. Marty, "Recherches sur la repartition des valeurs d' une fonction meromorphe", Ann. Fac.
Sc~.
Univ. Toulouse 23 (1931) 183-261.
24. H. Milloux, "Les cercles de remplissage des fonctions meromorphes ou entieres et Ie theoreme de Picard-Borel" , Acta
Mathemat~ca
52 (1928).
25. C. Miranda, "Sur un nouveau critere de normalite pour les families de fonctions holomorphes", Bullehn Soc. Math. 63 (1935) 185-196.
26. P. Montel, Lecons sur les
Fam~lles
Normales de
Fonct~ons Analllt~ques
et
leurs App hcatwns, Gauthier-Villars, Paris (1927).
27. P. Montel, Lecons sur les Fonctions U nwalentes ou Multwalentes, Gauthiers-Villars, Paris (1933).
28. R. Nevanlinna, Le Theoreme de
P~card-Borel
et la Theone des Fonctwns
Meromorphes, Gauthier-Villars ,Paris (1929).
29. W. F. Osgood, "Note on the functions defined by infinite series whose terms are analytic functions of a complex variable", Annals of Math. 3
(1901). 30. A. Ostrowski, "Veber Folgen analytischer Funcktionen und einige Verscharfungen des Picardschen Satzes", Math. Ze~t 24 (1925). 31. F. Schottky, "V eber den Picardschen Satz und die Borelschen V ngleichungen", S~tz.Kg.Akad., Berlin (1940).
467
References 32. W. Schwick, "Normality criteria for families of meromorphic functions" , Journal d' Analyse Math. 52 (1989) 241-289. 33. T. J. Steiltjes, "Recherches sur les fractions continues", Ann. Fac. SCl. Univ. Toulouse 8 (1894). 34. G. Valiron, "Sur les fonctions meromorphes qui admettent des valeurs quasi-exceptionnelles ," Assoc. fro avo
80. ,
(1926).
35. G. Valiron, "Sur un theoreme de MM. Koebe et Landau" , Bull. des Sc. math. 51 (1927) 34-42. 36. G. Valiron,
Familles Normales et
Quam-normales de
Fonctlons
Meromorphes, Memorial des Sc. Math. Fasc. 38 (1929). 37. G. Vitali, "Sopra Ie serie di funzioni analitiche", Rendlconti del R. 1st. Lombardo 36 (1903). 38. L. Yang, Value Dlstnbutwn Theory and
its New
Researches (in
Chinese), Science Press, Beijing, 1982. 39. L. Zalcman, "A heuristic principle in complex function theory", A mer. Math. Monthly 82 (1975) 813-817.
INDEX
( a)m-circle definition of,
333
Blaschke, W., 227 Blaschke's theorem, 232 Bloch, A., 40, 51, 52, 238 Bloch constant, 105, 241 Bloch's theorems, 51, 78 Bohr's theorem, 39 Caratheodory, C., 78 Chuang, C. T., 150 closed family of holomorphic functions definition of, 93 closed family of meromorphic functions definition of, 106 completely irreducible Cm-sequence in a domain definition of, 335 completely reduced sequence at a point definition of, 187 completely reduced sequence in a domain definition of, 194 condition of normality definition of, 136 condition of Qm-normality definition of, 330 condition of Qm-normality of order v at most definition of, 331 continuous functional definition of, 93, 107 continuous operator definition of, 95, 112 convergent with respect to the spherical distance definition of, 4 covering theorem, 123 Co-point definition of, 10
469
470
Index
C1-point definition Cm-point definition Co-sequence definition C1-sequence definition Cm-sequence definition
of,
131
of,
310
of,
10
of,
131
of,
310
equicontinuity with respect to the spherical distance definition of, 14 exceptional point of a sequence definition of, 222 exceptional value of weight 1 - ~ definition of, 77 filling circle, 37, 48, 71, 83, 120 filling circle in extended sense, 359 filling domain, 177, 178, 358 generalized (a)m-circle definition of, 351 Gu, Y. X., 445 Gu's theorem, 86 Hadamard, J., 403 Hayman, W. K., 86 irreducible C1-sequence with respect to a point definition of, 143 irreducible C1-sequence in a domain definition of, 143 irreducible Cm-sequence with respect to a point definition of, 334 irreducible Cm-sequence in a domain definition of, 334 Julia curve,
36
Index Julia direction, 35 Julia's theorem, 66
(K, e)-circle definition of,
137, 331
Landau, E., 40, 45, 55, 63, 122 Landau's theorem, 33, 103 limitation of modulus, 359 limitation of number of zeros, 209, 211, 379 limiting set of a sequence of sets definition of, 362 locally uniformly bounded in a domain definition of, 18 Macintyre, A. J., 393 Marty, F., 324 Marty's theorem, 24 (m)-collection of sets with respect to a domain definition of, 363 (m, v)-collection of sets with respect to a domain definition of, 363 method of Wiman-Valiron, 393 Miranda's theorem, 40 Montel, P., 78, 236, 353 Montel's condition at a point for a sequence definition of, 187 Montel's condition in a domain for a sequence definition of, 195 Montel's theorems, 18, 27, 57, 149, 150, 219 Nevanlinna, R., 443, 452 nonCo-point definition of, 131 nonCm-point definition of, 310 normal family of merom orphic functions definition of, 13 normality at a point definition of, 13
471
472
Index
order of a completely reduced sequence at a point definition of, 188 Osgood's theorem, 227 Ostrowski's theorem, 16 Picard's theorems, 29, 31, 60 Property W m of a set with respect to a domain definition of, 311 Property W m.L' of a set with respect to a domain definition of, 348 Qm-normal family of meromorphic functions definition of, 312 Qm-normal of order v at most definition of, 320 quasi-normal family of merom orphic functions definition of, 131 quasi-normality at a point definition of, 132 quasi-normal of order v at most definition of, 136 quasi-normal of total order T at most definition of, 196 reducible C1-sequence with respect to a point definition of, 143 reducible Cm-sequence with respect to a point definition of, 334 Schottky, F., 45, 55, 63 Schottky's theorem, 32, 102, 118, 361 simply covered image domain definition of, 51 spherical circle definition of, 71 spherical derivative definition of, 21 spherical distance definition of, 2 Stieltjes-Vitali's theorem, 220
Index
total order of a completely reduced sequence in a domain definition of, 194 uniformly convergent with respect to the spherical distance definition of, 6 univalent and multivalent functions, 249, 303 Valiron, G., 241 Valiron's theorems, Yang, L.,
244, 248
445
Zalcman's theorem,
73
11m-point of a sequence definition of, 318
473
