This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
T. Let i and (f>2 are two asymptotically conformal conjugacies such that cf)~l o f o 4>\{z) = Xz 1 \ and <j>2 are two conjugacies such that (jdj-1 og o (j)i(z) = zn 0, define ipt(x) = t~Nip{x/t). A locally integrable function / on RN is said to be in the Hardy space H1(MJV) if the maximal function M(f)(x)
107 Theorem 5.2. Lettx = [0],t2 = [l],t 3 = [T],^ = [I + T] <= R\R. Then the images J{tj)i3 = 1, 2 , . . . , 4, are represened by the Riemann surfaces Stj with one node: (1) Stl has two components isomorphic to the torus T, (2) St2 is the algebraic curve defiend by
y
~X
[X
0(1/2,T*)){X
0(0, r ) 2 ) '
(3) St3 is the algebraic curve defiend by v2 - x2 (x2 -
"
l
e
^r_mil_\
(x2 _
SWVFJl1
0{r/2,rf\
0(0, r)2 ) '
(4) Sti is the algebraic curve defiend by V2 - x2 (x2 + c - ^ ( l / 2 , r ) 2 \ (
2
_
6(r/2,r)2\
Moreover, ifreH satisfies - 1 / 2 < R e r < 0, \T\ > 1, or 0 < R e r < 1/2, \r\ > 1, then it follows that (1) J(t2) = J(t3) if and only if r = i,e™/ 3 , or (1 + \/l5i)/ 4 > (2) J(t2) = J(U) if and only if r = e7™/3, and (3) J(t 3 ) = J{U) if and only if r = e " / 3 , (1 + \/l5i)/4, or (1 + V7i)/2. The first part of Theorem 8 is prove by the explict representations for ft, St as (3), (4). In order to show the second part, we recall the universal covering A: H -> C \ {0,1, oo} defmded by A M _ A(T)
-
e
W0(T/2,r)\4 \e(i/2,T)) •
Then the second part is proved by using the following facts (cf. Clemens [2]): (1) by the blowing up at (x,y) = (0,0), the normaliztions of St2,St3,a.ad Stt are isomorphic to elliptic curves defined by y2 = (x — l)(x — A(2T)), y2 = x(x - l)(x - A(r/2)), and y2 = (x - l)(x - A(l/2 + T / 2 ) ) , respectively, (2) the universal covering transformation group of the universal covering A: H —> C \ {0,l,oo} is a Puchsian gropup generated by two elements T H T + 2 and r 1-+ T / ( 2 T + 1), which is a subgroup of PSL(2, Z) of index 6, and (3) for any two points T\,T2 6 H, elliptic curves defined by y2 = x(x — l)(x — A(ri)) and y2 = X(X — 1)(X — \(T2)) are isomorphic if and only if there exists an element 7 e PSL(2, Z) satisfying T2 = 7(ri).
108
References 1. S. Arakelov, Families of algebraic curves with fixed degeneracies, Math. USSR Izv. 35 (1971):1269-1293. 2. C.H. Clemens, A Scrapbook of Complex Curve Theory, Plenum Press, New York and London, 1980. 3. G. Faltings, Endlichkeitssatze fur abelsche Varietaten iiber Zahlkorpern, Invent. Math. 73(1983):349-366. 4. H. Grauert, Mordells Vermutung iiber rationale Punkte auf algebraischen Kurven und Funktionenkorper, IHES Publ. Math. 25(1965):131-149. 5. Y. Imayoshi, Holomorphic families of Riemann surfaces and Teichmuller spaces, in Riemann Surfaces and Related Topics, Stony Brook Conference, Ann. Math. Studies, 97(1978):277-300, edited by I. Kra and B. Maskit, Princeton University Press, Princeton, New Jersey, 1981. 6. Y. Imayoshi and H. Shiga, A finiteness theorem of holomorphic families of Riemann surfaces, in Holomorphic Functions and Moduli, Vol. II, pp.207-219, edited by D. Drain et al, Springer-Verlag:MSRI Publications Vol.10, 1988. 7. J. Jost and S-T. Yau, Harmonic mappings and algebraic varieties over function fields, Amer. J. of Math. 115 (1993):1197-1227. 8. K. Kodaira, A certain type of irregular algebraic surfaces, J. d'Analyse Math. 19 (1967):207-215. 9. Y. Manin, A proof of the analog of the Mordell conjecture for algebraic curves over function fields, Soviet Math. Dokl. 152(1963):1061-1063. 10. C. McMullen, From dynamics on surfaces to rational points on curves, Bull. Amer. Math. Soc. 37 (2000):119-140. 11. A.N. Parshin, Albegraic curves over function fields, Soviet Math. Dokl. 183 (1968): 524526. 12. G. Riera, Semi-direct products of Fuchsian groups and uniformization, Duke Math. J. 44 (1977):291-304.
a-ASYMPTOTICALLY CONFORMAL FIXED POINTS A N D HOLOMORPHIC MOTIONS*
YUNPING JIANG Department of Mathematics College of the City University of New York Flushing, NY 11367-1597 and Department of Mathematics Graduate School of the City University of New York 365 Fifth Avenue, New York, NY 10016 and Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100080 P. R. China E-mail: Yunping. Jiang @qc. cuny. edu Queens
An a-asymptotically conformal fixed point for a quasiconformal map defined on a domain is defined for any 0 < a < 1. Furthermore, we prove that for a > 0, there is a normal form for this kind attracting or repelling or super-attracting fixed point under quasiconformal changes of coordinate which is also asymptotically conformal at this point. These results generalize Konig's Theorem and Bottcher's Theorem in classical complex analysis. The idea in proofs is new and uses holomorphic motion theory and provides a new understanding of the inside mechanism of these two famous theorems.
Keywords: a-asymptotically conformal fixed point, holomorphic motion, normal form. 2000 Mathematics
Subject Classification:
Primary 37F99, Secondary 32H02
' T h e research is partially supported by NSF grants and PSC-CUNY awards and Hundred Talents Program from Academia Sinica. 109
110
1. Introduction Two of the fundamental theorems in complex dynamical systems are Konig's Theorem and Botthcher's Theorem in classical complex analysis which were proved back to 1884 19 and 1904 8 , respectively, by using some well-known methods in complex analysis. These theorems say that an attracting or repelling or super-attracting fixed point of an analytic map can be written into a normal form under suitable conformal changes of coordinate. These theorems become two fundamental results in the recent study of the dynamics of a polynomial or a rational map. However, it becomes more and more clear in recent years that only conformal change of coordinate is not enough in the study of many problems in dynamics and in geometry, for examples, in the study of monotonicity of the entropy function for the family \x\3 + t 29 and in the study of quasiconformal structures on a 4-manifold 1 0 -in these studies, quasiconformal changes of coordinate are appealed. The quasiconformal changes of coordinate may still have asymptotical conformality property just at one point but definitely not conformal. (It is a big difference between asymptotically conformal and conformal, see definition in Section 2.) During the study of complex dynamical systems, a subject called holomorphic motions becomes more and more interesting and useful. The subject of holomorphic motions over the open unit disk shows some interesting connections between classical complex analysis and problems on moduli. This subject even becomes an interesting branch in complex analo g 4,6,14,27,31,35 In this paper, we will use holomorphic motions over the open unit disk to study the quasiconformal changes of coordinate which are aymptotically conformal at one point. The paper is organized as follows. In Section 2, we give an overview about holomorphic motions and quasiconformal theory. In Section 3, we define an asymptotically conformal fixed point. We then definite an attracting or repelling a-asymptotically conformal fixed point. In Section 4, we prove one of our main theorems in this paper: Theorem 1.1. Let f be a quasiconformal homeomorphism defined on a neighborhood about 0. Suppose 0 is an attracting or repelling aasymptotically conformal fixed point of f for some 0 < a < 1. Then there is a quasiconformal homeomorphism
z £ As.
Ill
The conjugacy
1
is unique up to multiplication of constants.
To present our idea clearly, we first use the same idea in the proof of above theorem to give a new proof of Konig's Theorem in classical complex analysis in Section 3. Then we prove Theorem 1.1 in the same section. We definite an asymptotically conformal super-attracting fixed point in Section 3. In Section 5, we prove the other main theorem in this paper: Theorem 1.2. Let g — f(zn) be a quasiregular map defined on a neighborhood about 0. Suppose 0 is a super-attracting a-asymptotically conformal fixed point of g for some 0 < a < 1. Then there is a quasiconformal homeomorphism <j>: A<5 —> (j)(As) C U from an open disk of radius S > 0 centered at 0 into U which is asymptotically conformal at 0 such that (j)-1 o g o (j)(z) = Zn,
Z G AS.
x
The conjugacy 4>~ is unique up to multiplication by (n — l)th-roots unit.
of the
Again, we will first give a new proof of Bottecher's Theorem in classical complex analysis in Section 5. Then we prove Theorem 1.2 in the same section. 2. Holomorphic Motions and Quasiconformal Maps In the study of complex analysis, the measurable Riemann mapping theorem plays an important role. Consider the Riemann sphere C. A measurable function /J, on C is called a Beltrami coefficient if there is a constant 0 < k < 1 such that ||/i||oo < k, where || - ||oo means the L°°-norm of /x on C. The equation HY = iiHz is called the Beltrami equation with the given Beltrami coefficient fi. The measurable Riemann mapping theorem says that the Beltrami equation has a solution H which is a quasiconformal homeomorphism of C whose quasiconformal dilatation is less than or equal to K = (1 + k)/{\ - k). The study of the measurable Riemann mapping theorem has a long history since Gauss considered in 1820's the connection with the problem of finding isothermal coordinates for a given surface. As early as 1938, Morrey 32 systematically studied homeomorphic L 2 -solutions of the Beltrami equation (see 2 4 - 2 5 ). But it took almost twenty years until in 1957 Bers 5 observed that these solutions are quasiconformal (refer to 2 0 ) . Finally the existence
112
of a solution to the Beltrami equation under the most general possible circumstance, namely, for measurable fi with ||/u||oo < 1, was shown by Bojarski 7 . In this generality the existence theorem is sometimes called the measurable Riemann mapping theorem (refer to 1 6 ). If one only considers a normalized solution in the Beltrami equation (a solution fixes 0, 1, and oo), then H is unique, which is denoted as H*. The solution H11 is expressed as a power series made up of compositions of singular integral operators applied to the Beltrami equation on the Riemann sphere. In this expression, if one considers fi as a variable, then the solution ifM depends on /j, analytically. This analytic dependence was emphasized by Ahlfors and Bers in their 1960 paper 2 and is essential in determining a complex structure for Teichmuller space (refer to M6,20,26,33^ Note that when /i = 0, H° is the identity map. A 1-quasi-conformal map is conformal. Twenty years later, due to the development of complex dynamics, this analytic dependence presents an even more interesting phenomenon called holomorphic motions as follows. Let A r = {c S C | |c| < r} be the disk centered at 0 and of radius r > 0. In particular, we use A to denote the unit disk. Given a Beltrami coefficient u s£ 0, consider a family of Beltrami coefficients cu/llulloo for c £ A and the family of normalized solutions i? iiMiioo. Note that ffn*»ii°o is a quasiconformal homeomorphism whose quasiconformal dilatation is less than or equal to (1 + |c|)/(l — lc|). Moreover, JJiiiToo is a family which is holomorphic on c. Consider a subset E of C and its image Ec = .ffiiMiioo (£?). One can see that Ec moves holomorphically in C when c moves in A. That is, for any point z € E, z(c) = .ffiiciioo (2) traces a holomorphic path starting from z as c moves in the unit disk. Although E may start out as smooth as a circle and although the points of E move holomorphically, Ec can be an interesting fractal with fractional Hausdorff dimension for every c / 0 (see 1 5 ). Surprisingly, the converse of the above fact is true too. This starts from the famous A-lemma of Maiie, Sad, and Sullivan 28 in complex dynamical systems. Let us start to understand this fact by first defining holomorphic motions. Definition 2.1. [Holomorphic Motions] Let E be a subset ofC.
Let
h(c, z) : Ar x E -> C be a map. Then h is called a holomorphic motion of E parametrized by A r if
113
(1) h(0, z) = zforz€ E; (2) for any fixed c £ A r , h(c, •) : E —> C is injective; (3) for any fixed z, h(-, z) : A r —> C is holomorphic. For example, for a given Beltrami coefficient fi, H(c, z) = Wfcft
(z) : A x C -» C
is a holomorphic motion of C parametrized by A. Note that even continuity does not directly enter into the definition; the only restriction is in the c direction. However, continuity is a consequence of the hypotheses from the proof of the A-lemma of Marie, Sad, and Sullivan 28 . Moreover, Mane, Sad, and Sullivan prove in 28 that Lemma 2.1. [X-Lemma] A holomorphic motion of a set E c C parametrized by A can be extended to a holomorphic motion of the closure of E parametrized by the same A. Furthermore, Mane, Sad, and Sullivan show in 28 that /(c, •) satisfies the Pesin property. In particular, when the closure of E is a domain, this property can be described as the quasiconformal property. A further study of this quasiconformal property is given by Sullivan and Thurston 3 5 and Bers and Royden 6 . In 35 , Sullivan and Thurston prove that there is a universal constant a > 0 such that any holomorphic motion of any set E C C parametrized by the open unit disk A can be extended to a holomorphic motion of C parametrized by A a . In 6 , Bers and Royden show, by using classical Teichmuller theory, that this constant actually can be taken to be 1/3. Moreover, in the same paper, Bers and Royden show that in any holomorphic motion H(c, z) : A x C —> C, H(c, •) : C —> C is a quasiconformal homeomorphism whose quasiconformal dilatation less than or equal to (1 + |c|)/(l - |c|) for c e A. In the both papers 3 5 , 6 , they expect o = l. This was eventually proved by Slodkowski in 3 4 . Theorem 2.1. [Slodkowski's Theorem] Suppose h(c,z)
:AxE-^C
is a holomorphic motion of a set E C C parametrized by A . Then h can be extended to a holomorphic motion H(c, z) : A x C -> C
114
of C parametrized by also A. Moreover, following 6 , for every c € A, H(c, •) : C —> C is a quasiconformal homeomorphism whose quasiconformal dilatation
*(if(C,.))
Holomorphic motions of a set E C C parametrized by a connected complex manifold with a base point can be also defined. They have many interesting relationships with the Teichmuller space T(E) of a closed set E (refer to 3 1 ) . In addition to the references we mentioned above, there is a partial list of references 3,4,11,12,13,14,27,21,22 a b o u t holomorphic motions and Teichmuller theory. The reader who is interested in holomorphic motions may refer to those papers and books. 3. a-Asymptotically conformal fixed points We define a class of quasiconformal maps and quasiregular maps fixing a point. Let / be a quasiconformal homeomorphism defined on a neighborhood U about 0 and fixing 0. Let n{z) = fzlfz be the complex dilatation of / on U. For any t > 0 such that A t C U, let u(t) = \\fj,\At\\oo, where || • ||oo means the L°° norm. Definition 3.1. We call / asymptotically conformal at 0 if w(t) -> 0 as
t -> 0 + .
Furthermore, for a real number 0 < a < 1, we call / a-asymptotically conformal at 0 if lim ta f ^r-ds = 0, t-o+ Jo « 1 + Q If / is asymptotically conformal at 0, then / maps a tiny circle centered at 0 to an ellipse and, moreover, the ratio of the long axis and the short axis tends to 1 as the radius of the tiny circle tends to 0. But the map still can fail to be differentiable at 0 (refer to 1 7 ). However, following Reshetnyak's 1978 paper 2 3 ) , if / is O-asymptotically conformal at 0, then / is differentiable and conformal at 0, i.e., the limit of f(z)/z exists as z goes to 0. Let
A=hmM |z|-»0
and call it the multiplier of / at 0.
Z
115
It is clear that for 0 < a' < a < 1, if / is a-asymptotically conformal at 0, then it must be a'-asymptotically conformal at 0. Therefore, if / is a-asymptotically conformal at 0, then one can define its multiplier A at 0. We call 0 i) attracting if 0 < |A| < 1; ii) repelling if |A| > 1; hi) neutral if |A| = 1. Correspondingly, we call 0 an attracting, repelling, or neutral aasymptotically conformal fixed point. Let g be a quasiregular map defined on a neighborhood U about 0 and fixing 0. Assume g = f o qn where qn(z) = zn and / is a quasiconformal homeomorphism. We say g is a-asymptotically conformal at 0 if / is aasymptotically conformal at 0 with nonzero multiplier A = lim| z |_ 0 f(z)/zIn this case 0 is called a super-attracting a-asymptotically conformal fixed point of g. The following lemma will be useful in our proofs of Theorems 1 and 2. Lemma 3 . 1 . Suppose u)(t) is an increasing function oft > 0. Suppose, for some 0 < a < 1,
Um f f ^Lds = 0. Suppose 0 < a < 1. Let oo
^2a-anuj{ant).
Q(t) = j=n
Then
w
-
-logWo
*1+Q
Moreover, w(t) —» 0 as t —* 0 + . Proof. Since w(t) is increasing for t > 0, we have u,(t) < r
Ji
a-axtj(oxt)dx
= -^—
f7
-logWo
^4r-ds.
« 1+Q
n
D
116
4. Linearization for a-asymptotically conformal attracting or repelling fixed points One of the main results in this article is Theorem 1, which says that if / is a quasiconformal homeomorphism and 0 is an attracting or repelling a-asymptotically conformal fixed point with the multiplier A, 0 < |A| < 1 or |A| > 1, and with a > 0, then / can be written as a linear map z —> Xz under some quasiconformal change of coordinate which is also asymptotically conformal at 0. The result generalizes the famous Konig's Theorem in classical analysis. Therefore, to present a clear idea about how we get Theorem 1, we first use the same idea to give another proof of Konig's Theorem. The idea of the new proof follows the viewpoint of holomorphic motions. For the classical proof of Konig's Theorem, the reader may refer to Konig's original paper 19 or most recent books 9 ' 30 . Actually from the technical point of views, our proof is more complicate and uses a sophistical result. But from the conceptual point of views, our proof gives some inside mechanism for the linearization of an attracting or a repelling fixed point. Theorem 4.1. [Konig's Theorem] Let f(z) = Xz + Y^?=2aiz^ ^e an an~ alytic function defined on Aro, TQ > 0. Suppose 0 < |A| < 1 or \X\ > 1. Then there is a conformal map <j> : As —>
zeAs
and / is injective on A,*. For every 0 < r < S, let Sr = {z G C I \z\ = r} and Tr = |A|Sr = {z G C | \z\ = \X\r}. Denote E = Srl)Tr.
Define ( Z Mz)
=
Z&Sr
\f(i),zeTr.
117
It is clear that fa1
° /'0
for z £ Sr. Now write <pr(z) = zipr{z) for z £ Tr, where oo
J'=l
Define
:A ,e
' ~t
**•> = {*<*•),"£ Note that
««„_,*(2«)_^(2«) = £,(£), .cr.co. For each fixed z £ E, it is clear that h(c, z) is a holomorphic function of c £ A. For each fixed c £ A, the restriction h(c, •) to Sr and Tr, respectively, are injective. Now we claim that their images do not cross either. That is because for any z £ Tr, \z\ = |A|r and |cz£|/|rA| < 6, so \ui
M
r
\\t(czS\\
^
r
c z S
Therefore, h(c, z) : A x E —> C is a holomorphic motion because we also have h(0, z) = z for all z E E. From Theorem 3, h can be extended to a holomorphic motion H(c, z) : A x C - t C, and moreover, for each fixed c & A, Hc = h(c,-) : C —> C is a quasiconformal homeomorphism whose quasiconformal dilatation is less than or equal to (1 + |c|)/(l — |c|). Now take c r = r/S and consider H(cr,-). We have H(cr, -)\E = <j>r. Let ArJ
= {z £ C | |A| J + 1 r < |z| < |A| J 'r}.
We still use (f>r to denote H(cr, -)\Ario. For an integer k > 0, take r = r/c = 5\\\k. Then A* = U°i_ f c A r J U {0}. Extend
= rj(M>rz)),
zGAr,j,
i = -fc,-..,-l,0,l,---,
and
118
dilatation is the same as that of H(cr,-) on ATtQ. So the quasiconformal dilatation of <j>r on As is less than or equal to (1 + r ) / ( l — r). Furthermore, f{(j>r{z)) = 4>r{Xz),
zeAj.
Since f(z) = Az(l + O(z)), fk(z) = XkzUi^0+ O(A'z)). Because |A| r/t = S, the range of <j)rk on A^ is a Jordan domain bounded above from oo and below from 0 uniformly on k. In addition, 0 is fixed by <j>k and the quasiconformal dilatations of the <j>k are uniformly bounded. Therefore, the sequence {
f(cf>(z)) = <}>(\z),
Z
£A,
The quasiconformal dilatation of <j> is less than or equal to (1 + rfc)/(l — r^) for all k > 0. So 0 is a 1-quasiconformal map, and thus is conformal. This is the proof of the existence. For the sake of completeness, we also provide the proof of uniqueness but this is not new and the reader can find it on 9 ' 3 0 . Suppose i and
and
(f)^1 ° / o(j>2(z) = Xz,
z £ As.
Then for $ =
|/(z)|<|z|,
zeAgcU
and / is injective on A^. For every 0 < r < S, let Sr = {z G C \\z\ = r} and Tr = aSr = {z e C | \z\ = ar}.
119
Denote E = SrUTr.
Define .
( Z
Z G Sr
zeTr. It is clear that
Oiwia-'r)).
Consider uc = caoa(r)~lv and the unique solution
= { z £ C | aj+1r < \z\ <
We still use
ZCArj,
j = - A , • • • , - 1 , 0, 1, • • • ,
and 4>r(0) = 0. Since 4>r\E is a conjugacy from / to Xz, 4>T is continuous on A<5. Next we need to estimate the quasiconformal constant of cf>r on Ag. We will use the following formula (refer to 1): If F and G are two quasiconformal
120
maps with the complex dilatations fip and /XG- Then the composition map G o F has the complex dilatation fiF + 7MG ° F MGoF = T-— =, 1 + MF7MG °F
where
FT 7 = —. FZ
Thus IIMGOFIIOO
<
+
(IIMF||OO
IIMG
° ^lloo)(i -
IIMFII OO||MG
° -flloo)
Suppose, in the beginning of the proof, we pick 5 small such that (l-u>(6)\\l*g°f\\«>r1<
||/i^||oc < J2°~aJ"(°JS) + b(r) ^ "(*) + b(r) and \x^T over A r can be controlled by 00
||/V r |Ar||oc < Y^a~ai^T)
+ 6 ( r ) ^ "(r) + b(r)-
i=i
Thus {4>r}o
zeAs.
Since the complex dilatation of cfi is controlled by b(r) = u>(r) + b(r),
and
cp^1 ° f ° $2(2) = Xz,
z € As.
Then for $ = cj)^ ° 4>i, we have $(\z) = A$(z). This implies that the complex dilatation /i$(z) = /J,$(\Z), a.e. This in turn implies that /i = 0
121
a.e. in As and thus $ is conformal. Furthermore, $(z) = az for some a ^ O . This is the uniqueness. •
5. Normal forms for a-asymptotically conformal super-attracting fixed points The other main result in this article is Theorem 1.2, which says that if g = f{zn) is a quasiregular map and 0 is an a-asymptotically conformal superattracting fixed point, then g can be written into the normal form z :—> zn under some quasiconformal change of coordinate which is asymptotically conformal at 0. The result generalizes the famous Bottcher's Theorem in classical analysis. Again, to present a clear idea about how we get Theorem 1.2, we first use the same idea to give another proof of Bottcher's Theorem. The idea of the new proof follows the viewpoint of holomorphic motions. For the classical proof of Bottcher's Theorem, the reader may refer to Bottcher's original paper 8 or most recent books 9 ' 30 . Actually from the technical point of views, our proof is more complicate and uses a sophistical result. But from the conceptual point of views, our proof gives some inside mechanism of the normal form for a super-attracting fixed point. The idea of the proof is basically the same as that in the previous section, but the actual proof is little bit different. The reason is that in the previous case, / is a homeomorphism so we can iterate both forward and backward, but in Theorem 1.2or Bottcher's Theorem, g is not a homeomorphism. Theorem 5.1. [Bottcher's Theorem] Suppose g(z) = Y^LnaJz'' > a™ ^ ®> n > 2, is analytic on a disk As0, SQ > 0. Then there exists a conformal map <j>: As —>
zeAs.
The conjugacy
of the
Proof. [A new proof of Bottcher's Theorem] Conjugating by z —• bz, we can assume an = 1, i.e., oo
g(z) = zn+
Yl j=n+l
a zJ
o -
122
We use A* = A r \ {0} to mean a punctured disk of radius r > 0. Write oo
g(z) = zn(l +
J2aj+nzj). 3= 1
Assume 0 < Si < min{l/2, J 0 /2} is small enough such that •
°°
1 + ^2aj+nZJ
1
+0
and
1
^ ^^ > -, \/\l + Z7=laJ+nZi\
j=i
z'e
A2Sl.
Then g : A^Sl —> / ( A ^ ) is a covering map of degree n. Let 0 < 5 < Si be a fixed number such that 5-1(A<5) C A ^ . Since z^zn:A*^->A*s
and fl : fl-^AJ)-» AJ
are both of covering maps of degree n, the identity map of As can be lifted to a holomorphic diffeomorphism h:A*^-^g-\A*5), i.e.,, h is a map such that the diagram
A* ,-
3' n
J.z-> z
Is id>
AS
AS
commutes. We pick the lift so that oo
h(z) = z(l + Y,bizJ~l)
=
z z
^ )'
j=2
From g(h(z)) = zn,
Z
£A^,
we get
|M*)| = ,
'*'
For any
0
> f.
123
let Sr = {z £ C | \z\ = r} and Tr = {z £ C | \z\ = tfr}. Consider the set E = Sr U Tr and the map z,
Mz) = \'ztp(z),
z £ Sr z e Tr.
Define Z)
K{c,z) = l
Z \i
(^_\
by
^cT
:Ax£-.C.
Note that
This implies that ~
c 2{/F
_
2
So images of SV and T r under /i r (c, z) do not cross each other. Now let us check hr(c,z) is a holomorphic motion. First hr(0,z) = z for z £ E. For fixed x £ E, hr(c,z) is holomorphic on c £ A. For fixed c £ A, hr(c,z) restricted to Sr and T r , respectively, are injective. But the images of Sr and Tr under hr(c,z) do not cross each other. So hr(c,z) is injective on E. Thus /i r (c,z) : A x £ ^ C is a holomorphic motion. By Theorem 2.1, it can be extended to a holomorphic motion Hr(c,z)
:AxC-.C.
And moreover, for each c £ A, Hr(c,-) quasiconformal dilatation satisfies
is a quasiconformal map whose
*(*(*,»< i±M Now consider H( $/F, •). It is a quasiconformal map with quasiconformal constant
1 - tfr Let A r j = { z £ C | " v ^ < | z | < "J+v^},
j=0,l,2,....
124
Consider the restriction
A
A
I z - *"
I 9 >
-™T,0
A-rfl
commutes. We pick the lift <j>rt\ such that it agrees with cj>rfl on Tr. The quasiconformal dilatation of ri is less than or equal to Kr. k
For an integer k > 0, take r = r^ = 6n . Inductively, we can define a sequence of .Kr-quasiconformal maps {(j>r,j}j=o s u c n that A
•
^H
i z - i z"
A
•
|ff
commutes and t/vj and
«*>={*:, z € A ; r
-j,
Z fc ^r,j'i J
=
" i •!•! ' ' ' > "'•
The quasiconformal dilatation of <pr on A^ is less than or equal to Kr and g{4>r{z)) = <j>r{zn), z G U^ =1 Ar Since g(z) = z n ( l + O(z)), 5 fc (z) = zn YUZQ^ + 0{zn%)). Because -J/rfc = 6, the range of rfc on A<s is a Jordan domain bounded above from oo and below from 0 uniformly in k. In addition, 0 is fixed by (j>k and the quasiconformal dilatations of the <j>k are uniformly bounded in k. Therefore, the sequence {rfc}fcLi is a compact family (see 1 ) . Let 0 be a limiting map of this family. Then we have n
f(
zeAs.
125
Since the quasiconformal dilatation of is less than or equal to (1 + \/r f e )/(l — \/rk) for all k > 0, it follows that
and
(j)^1 o g o
z € As.
For oo
$(z) = < ^ o0i(z) = 5 3 Oj^', we have $(zn) — ($(2))". This implies a j = ax and a^ = 0 for j > 2. Since ai 7^ 0, we have a " - 1 = 1 and cj)^1 = ai^f 1 - This is the uniqueness. • We now prove Theorem 1.2. The proof follows almost the same footsteps of those of Theorem 1.1 and Theorem 5.1. Proof. [Proof of Theorem 1.2.] Let g = f o qn, n > 2. Conjugating by 2 —> bz, we can assume /'(0) = lim| z |_ 0 f(z)/z = 1We use A* = A r \ {0} to mean a punctured disk of radius r > 0. There is a 0 < 5\ < 1 such that g : A%s —> / ( A ^ ) is a covering map of degree n. Let 0 < S < Si be a fixed number such that g~1(As) C A ^ . Since
*-*n:A*^->A;
and «, : ^ ( A J ) - » AJ
are both of covering maps of degree n, the identity map of A<5 can be lifted to a homeomorphism
h:Ay5^g-\A*s). Furthermore, ft is a quasiconformal map a-asymptotically conformal at 0 such that the diagram
lz^zn
A*
lg
ii
A*
commutes. We pick the lift so that
ti(z) = lim Mf) = 1.
126
These can be seen from the equation g(h(z)) = zn,
Z
£A^.
For any 0 < r < 6, let Sr = {z G C | \z\ = r} and Tr = {z G C | \z\ \/r}. Consider the set E = SrUTr and the map .(z) = |
Z,
Z G Sr
h(z), z G Tr.
It is clear that
g{M*)) = M*n) for z G Sr. Extend h to C by quasiconformal reflection with respect to Sr and 4>(Sr) (see 1 ) . We still denote this extended map as
<pr(c,z) = I
It is a holomorphic motion from A x E —> C. From Theorem 2.1, <j>(c,z) can be extended to a holomorphic motion from A x C —> C, which we still denote by (f>(c,z), such that the quasiconformal dilatation of
=
ArJ
= {zeC|
n
^/F< \z\ <
ni+
V^}-
We still use 4>r to denote <j>r{cr, OlA-.o- For an integer k > 0, take r = rk = n
!/6. Then A^ = U?=-kAr,j U {0}.
127
Extend
z e A
r J
,
j = -k,
• • • , - 1 , 0, 1, • • • ,
and <j>r(0) = 0. Since <pr\E is a conjugacy from g to qn{z) = zn, <j)r is continuous on A<5. Using the similar argument to that in the proof of Theorem 1.1, we can get some 0 < a < 1 such that the complex dilatation /x^r over As can be controlled by oo
* " a M nVS) + Kr) < Q(S) + b(r)
IK,lloo < £
and \i$r over A r can be controlled by oo
H/x^lArlloo < £ a - a j w ( nV6) + b{r) < u{r) + b(r). Thus {
ZGAS.
Since the complex dilatation of
= z"
and
0 J 1 og o(j)2{z) = zn,
ZGAS.
Then for $ = (j)^1 o <j>t, we have $(2") = ($(z))n. This implies that the complex dilatation ||//$(z)|| = ||/x$(z n )||, a.e. This in turn implies that (j, = 0 a.e. in Aa and thus $ is conformal, and therefore, $(z) = az with an = 1. This is the uniqueness. •
128
A c k n o w l e d g m e n t . T h e previous version of this paper 1 8 was first discussed in the complex analysis and dynamical systems seminar at the C U N Y G r a d u a t e Center in the Spring Semester of 2003. I would like to t h a n k all participants for their patience. During the discussion Professors Linda Keen, Fred Gardiner, a n d Nikola Lakic provided m a n y useful comments to improve and polish this article. Sudeb Mitra explained to me several points in the development of the measurable Riemann mapping theorem and holomorphic motions. I express my sincere thanks to them. I would like also to t h a n k Professor Weiyuan Qiu to help me to clarify several arguments in this paper.
References 1. L. V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand Mathematical studies, 10, D. Van Nostrand Co. Inc., Toronto-New York-London, 1966. 2. L. V. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Annals of Math. (2), 72 (1960): 385-404. 3. K. Astala, Planar quasiconformal mappings; deformations and interactions, In Quasiconformal Mappings and Analysis-A collection of papers honoring F. W. Gehring, 1998, Springer-Verlag, New York, Inc., 33-54. 4. K. Astala and G. J. Martin, Holomorphic motions, Papers on Analysis, 27-40, Rep. Univ. Jyvaskla Dep. Math. Stat., 83, Univ. Jyvaskyla, 2001. 5. L. Bers, On a theorem of Mori and the definition of quasiconformality, Trans. Amer. Math. Soc, 84 (1957): 78-84. 6. L. Bers and H. L. Royden, Holomorphic families of injections, Acta Math, 157 (1986): 259-286. 7. V. B. Bojarski, Generalized solutions of a system of differential equations of the first order and elliptic type with discontinuous coefficients, Math. Sbornik, 85 (1957): 451-503. 8. L. E. Bottcher, The principal laws of convergence of iterates and their aplication to analysis (Russian), Izv. Kazan. Fiz.-Mat. Obshch. 14) (1904):155-234. 9. L. Carleson and T. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993 10. S. K. Donaldson and D. P. Sullivan, Quasiconformal -manifolds, Acta Math. 163 (1989):181-252. 11. A. Douady, Prolongement de mouvements holomorphes [d'apres Slodkowski et autres], Seminarie N. Bourbaki (1993/1994), Asterique No. 227 (1995) & Exp. No. 755 (3): 7-20. 12. P Duren, J. Heinonen, B. Osgood, and B. Palka, Quasiconformal Mappings and Analysis, Springer, 1997 13. C. J. Earle, I. Kra, and S. L. Krushkal, Holomorphic motions and Teichmiiller spaces, Trans. Amer. Math. Soc, 343 (2) (1994):927-948. 14. C. J. Earle and S. Mitra, Variation of moduli under holomorphic motions, Comtemp. Math., 256 (2000): 39-67.
129
15. F. Gardiner and L. Keen, Holomorphic motions, Lipman Bers, Selected Works Part I and II, Edited by I. Kra and B. Maskit, AMS, 1999. 16. F. Gardiner and N. Lakic, Quasiconformal Teichmiiller Theory, Mathematicak Survey and Monographs, American Mathematical Society, Rhode Island, 76 (2000). 17. V. Ya. Gulyanskii, O. Martio, V. I. Ryazanov, and M. Vuorinen, On the asymptotic behavior of quasiconformal mappings in space in Quasiconformal Mappings and Analysis (edited by P Duren, J. Heinonen, B. Osgood, and B. Palka), Springer, 1997: 159-180. 18. Y. Jiang, Holomorphic Motions and Normal Forms in Complex Analysis, Proceesings of ICCM2004, to appear. 19. G. Konigs, Recherches sur les integrals de certains equations fonctionelles, Ann. Sci. Ec. Norm. Sup., (3 e ser.) 1 (1884): supplem. 1-41. 20. O. Lehto, Univalent Functions and Teichmiiller Spaces, Springer-Verlag, New York, Berlin, 1987. 21. C. McMullen and D. Sullivan, Quasiconformal Homeomorphisms and Dynamics III: The Teichmiiller space of a holomorphic dynamical systems, Adv. Math., 135 (1998):351-395. 22. Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, 1992. 23. Yu. G. Reshetnyak, Stability theorems in geometry and analysis, Nauka, Sibirskoe otdelenie, Novosibirsk, Russian, 1982. 24. M. A. Lavrent'ev, Sur une classe de representation continues, Mat. Sbnornik 42(1935):407-423. 25. M. A. Lavrent'ev, A fundamental theorem of the theory of quasi-conformal mapping of plane regions. Izvestya Akademii Nauk S.S.S.R. 12 (1948):513554. (Russian). 26. Z. Li, Quasiconformal mappings and applications in Riemann surfaces, Science publciation of China, 1988. 27. G. Lieb, Holomorphic motions and Teichmiiller space, Ph.D. dissertation, Cornell University, 1990. 28. R. Marie, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Ec. Norm. Sup., 96 (1983): 193-217. 29. W. de Melo and S. van Strien One-Dimensional Dynamics, Springer-Verlag, Berlin, New York, 1993. 30. J. Milnor, Dynamics in One Complex Variable, Introductory Lectures, Vieweg, 2nd Edition, 2000. 31. S. Mitra, Techmiiller spaces and holomorphic motions, Journal d'Analyse Mathematique, 81 (2000): 1-33. 32. C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Math. Soc, 43 (1938): 126-166. 33. S. Nag, The Complex Analytic Theory of Teichmiiller Spaces, John Wiley and Sons, New York, 1988. 34. Z. Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc, 111 (1991): 347-355. 35. D. Sullivan and W. Thurston, Extending holomorphic motions, Acta Math., 157 (1986):243-257.
T H E H Y P E R - O R D E R OF SOLUTIONS OF CERTAIN HIGH ORDER DIFFERENTIAL EQUATIONS*
CHUNHONG LI College of Mathematics and Physics Chongqing University Chongqing, 400044 P- R- China and College of Math, and Information China West Normal University Nanchong, 637002 P. R. China E-mail: [email protected] YONGXING GU College of Mathematics and Physics Chongqing University Chongqing, 400044 P- R- China E-mail: [email protected]
This present paper investigate the growth problem of solution of certain highorder linear differential equations and obtain precise estimates of hyper-order of their solutions with infinite order.
Keywords: Linear differential equations, hyper-order. 2000 Mathematics Subject Classification: 30D35.
In this paper, we will use the s t a n d a r d notations of the Nevanlinna theory(see e.g., [1,2,3]). In addition, we use cr(/),<7 2 (/) t o denote respectively the order of growth of f(z). •Foundation Term: NNSF of China No. 10271122; NNSF of the Education Department of Sichuan No. 2004al04. 130
131
J. K. Langley' 4 ', G. Gundersent5!, Z. X. Chen'6! etc. concerned the problem of growth of solutions of the second order differential equation
f"+e-zf'
+ Q(z)f = 0.
(1)
They proved the following [4,5,6] Theorem A Let Q{z) be nonconstant entire function and satisfy one of the following conditions (i) Q(z) be nonconstant polynomial; (ii) a(Q) < 1 oro(Q) > 1 ; (Hi) Q{z) = h(z)e~dz where h(z)(^ 0) be entire function and o(h) < l,d(^ 1) be a nonzero complex. Then all solution f(z)(^ 0) of (0.1) have infinite order. It is a very important aspect in the complex oscillation theory to more precise estimates for the rate of growth of infinite order solutions of linear differential equations in the complex domain I6'. Z. X. Chen proved the following Theorem B Let a be nonconstant polynomial, orQ(z) = h(z)e~dz, where h(z)(^ 0) be nonzero polynomial. Then all solution f(z)(^ 0) of (0.1) have infinite order and C2(/) = 1 where the hyper-order o-^if) of f(z) is defined byM cr 2 (/) = hmsup r—oo
loglogT(r,f) . ; logr
A nature problem is what qualities about the growth of solution of the high order differential equation corresponding to (0.1)? C. H. Lit8' a n c i x . J. Huang!8! researched the problem and proved that Theorem C Let Aj(z)(=£ 0) be entire functions with cr(Aj) < l(j = 0 , 1 , . . . , k - 1; k > 2), a,j € C\ {0}(j = 0 , 1 , . . . , k - 1) with a,j = Cja0 (.7 = 1 , . . . , A; —1), and that ifck-i > c = maxi<j
+•••+ A0(z)eaoZf
=0
(2)
have infinite order. In this present paper, we consider the hyper-order of infinite order solutions of linear differential equation (0.2) Aj(z) (j = 0 , 1 , . . . , k — 1) with being nonzero polynomials.
132
Theorem 1 Let Aj(z)(j = 0 , 1 , . . . , k — 1) be nonzero polynomials, aj € C \ {0}(i = 0 , 1 , . . . , fc — 1), Cj > l ( j = 1 , . . . , fc — 1), and that there exists m e { 1 , 2 , . . . , k — 1}, SMC/I £/iat cm > c = max.i<j
- ez f = 0
has a solution /(z) = z with nonzero and finite order. 1. L e m m a s Lemma lJ9l Let f(z) be an entire function of infinite order, with the hyper-order <72(/) = o~ < oo, and let u(r) be the central index of f(z). Then loglogv(r-) hmsup = a2(f) = a. ; r^oo logr Lemma 2™ Let f(z) be an entire function with a(f) = a < +oo, then for any e > 0, there exists a set EQ C (1, +oo) with finite linear measure mEo < +oo, and finite logarithmic measure ImEg < +oo, such that for z satisfying \z\ = r $ [0,1] U E0, |/(z)|<expK+£}. Lemma 3 Let Bj(z)(j = 0 , 1 , . . . , k — 1) be entire functions with finite order. If f(z) is any solution of infinite order of the equation /<*> + B f c - ^ z ) / ' * - 1 ' + • • • + Bo(z)f = 0
(3)
then a2(f) < max.0<j
= 0,1,...,k-1).
(4)
By Wiman-Valiron Theory, there exists a set E\ C (1, +00) having finite logarithmic measure lmE\ < +00, we can choose \z\ = r £ [0,1] \JE\ and \f(z)\ =M(r,f), we get J— = (^2y(i
+ 0(i))(j
=
i,2,...,k)
(5)
133
where v{r) is the central index of f{z). Prom (1.3), we have -~=Bk.xJ-~—
+ ... + £ '
+
B0.
(6)
Substituting (1.4)(1.5) into (1.6), for sufficiently large r(\z\ = r ^ [0,1] U ^ o U ^ i and |/(2)| = M(r, / ) ) , we have ( ^ ) f c ( l + 0 ( l ) ) < exp{r°+*}{C^)k-\l+o{l))+.
• .+(f^)(i+0(i))+i}
We can further get ( ^ < 4 e ^ K ^ \z\
{
(
^ ) \z\
+
. . . + (^) \z\
+
l}
(V(r)\k _ ^
4exp{r°+E}-^
,
(notice that a{f) = oo and let r 3> 1 ), i.e. v(r) < 5exp{ra+'}(r
> 1)
i.e.
lim sup r^[0,l]UBflU£i,r^oo
loglogvir) _ a 1 < cr + e. *-°9r
Since e is arbitrary, by Lemma 1.5 in [3] and Lemma 1, we have o~2(f) < a. Lemma 4^ Suppose that p(z) = (a + i(3)zn -\ (a, (3 e R, \a\ + \/3\ £ 0) is polynomial with degp(z) = n > 1, A(z)(j£ 0) is an entire function with a(A) < n, g(z) = A(z)exp{p(z)}, z = re%e, S(p,6) = acosnQ — fishinO. Then for any e > 0, there exists E-i c (1, +oo) with ImE^ < +oo,such that for any 6 e [0,2TT] \ Hx(Hi
= {6 e [0,2TT] : 5{p,0) = 0} is an finite set),
and \z\ = r $ [0,1] U E2, the following are true: (i) If 6(p, 6) > 0, we have exp{(l - e)S(p, e)rn) < \g(rei9)\ < exp{(l + e)5(p, 6)rn}; (ii) If 6(p,0) < 0, we have exp{(l + e)S(p,6)rn}
< \g(rei9)\ < exp{(l -
e)6(p,0)rn}.
134
Lemma 5l6l Let f(z) be an entire function with cr(f) = +oo,<72(/) = a < +00, and a set E3 C (l,+oo) with ImE^ < +00. Then there exists zi = rteie'(l = 1,2,...), with |/(z,)| = M{n,f), 6t e [0,2TT),lim z ^ +00 = #o S [0,2ir), ri <£ E3, ri —• +00, /or Ve > 0 ,and for sufficiently large ri, we have exp{rf
£
i:™™™^^) } < z/(r;) < exp{r,„
2. Proof of Theorem 1 By Theorem C, Lemma 3 and combining the conditions of Theorem 1, for all non-trivial solutions f(z) of the equation (0.2), we have <x(/) = 00,
/ ^ M = ( ^ ( l + 0 (l)),j = l,2,...,fc.
(7)
J\z) \z\ where \z\ = r <£ [0,1]U-Ei> l™Ei < + ° ° , \f(z)\ = M(r,f), u(r) central index of f(z). By Lemma 5, there exists zi = riel6l(l = 1,2,...), \f{zi)\ = M(rhf), 9i e [0,2TT), l i m ^ + 0 0 = 60 e [0,2TT), [0,1] |J E0 IJ £1 |J £ 2 U ^3> n -> +00, for Ve > 0 ,and for sufficiently ri, we have
is the with n £ large
limsuP^l=+oo, i^+00 log 77
(8)
e x p { r p £ } < v(n) < exp{rf + £ }.
(9)
For above #o , by the assumptions of Theorem 1, we may let a = r^e^ such that 5(aoz,0o) = rocos((j) + QQ) = 6 ^ 0 . Now we divide into the following two cases to prove: case(i)<5 < 0, case(ii)<5 > 0 . Case (i) 6 < 0. By lim/^+oo = #o> f° r sufficiently large I, we have S(aoz,6i) = o"; < 0, notice that a,j = CjOo(j = 1 , . . . , k—1), we can get S(a,jZ, 8{) = Cj6(aoz, Qfi = CjSi < 0, S(-a0z,9i) = -Si > 0,S(ajZ - a0z,6i) = S(a0(cj - l)z,6i) = (CJ - l)Si < 0. From (0.2), we have f(fc- 1 )
f(fc) _e-aQzJ
=
Ak_1^ea0(ck_1-l)zJ_^
+
f' .. . +
yl1(2)e«o(ci-l)«£.+i4o(z)-
(10)
135
(0,
Substituting (1.7)(1.9) into (1.10), and by Lemma 4, for any given e £ and for sufficiently large I , we obtain
—Lc ^~Y ), 2
exp{(l - e )(-*,)rj}exp{ATf- e }rr f c (l + o(l)) < | -
e~a<*f-^-\
< exp{(l -e)(c f c _i - l)<5(rjexp{(fc - l ) r f + £ } r r f c + 1 ( l + o(l)) + exp{(l - £)(cfc_2 - 1)<W exp{(fc - 2)rf + £ }r ; - f e + 2 (l + o(l)) + e x p { ( l - e ) ( c i -l)<5 ; r i }exp{rf + e }rf 1 (l + o ( l ) ) + r n + 1
(11)
where n = maxo<j
-Am{z)ea^J-j-
f(m)
f(k)
f(m-l)
= J— + ... +
Am.1(z)ea^zJ—y-
f(m+l)
+ Am+1{z)ea^zJ——
+ ••• + A0(z)eaoZ.
(12)
Substituting (1.7)(1.9) into (1.12), and by Lemma 4, we get e x p { ( l - e ) c m ^ r i } e x p { m r p £ } r i - m ( l + 0 (l)) < | -
Am{z)ea™zi—\
< exp{fcrf +e }rf*(1 + o(l)) + • • • + exp{(l + e)cTO_1(5(r(} x exp{(m - l)r ; C T + £ }r,- m + 1 (l + o(l)) + exp{(l +
e)cm+16iri}
x exp{(m + l ) r f + e } r f T n - 1 ( l + o(l)) exp{(l + e)<J,rj}.
(13)
Taking sufficiently small such that 0<2e<min{l-g,
Cm
~°},
(14)
Cm + C
and noticing that 0 < a < 1 , by (1.13) and (1.14), for sufficiently large I, we have e x p { ( C m ~ 2 C ^ ' r ' } < 4(ib - l)exp{jfcrf +e }r?\ This is impossible and the Theorem 1 follows.
136
References 1. W. Hayman,Meromorphic Function, Oxford: Clarendon Press, 1964. 2. L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993. 3. S.A. Gao, Z.X.Chen and T.W.Chen, The Complex Oscillation Theory of Linear Differential Equations (in Chinese), Huazhong University of Science and Techology Press, Wuhan, 1998. 4. J. K. Langley, On Complex Oscillation and a Problem of Ozawa, Kodai Math.J., 9(1986):430-439. 5. G. Gundersen, On the Question of Whether/" + e~zf + B(z)f = 0 Can Admit a Solution /(=£ 0) of Finite order, Proc. R. S. E., 102A(1986):9-17. 6. Z.X.Chen, The Growth of Solutions of Differential Equations/" + e~zf + Q(z)f = 0 (in Chinese), Science in China(ser. A), 31(9)(2001):775-784. 7. H.X.Yi, C.C.Yang, The Uniqueness Theory of Meromorphic Functions (in Chinese), Science Press,Beijing, 1998. 8. C.H. Li, X.J.Huang , The Growth of Solution of a Class of High Order Differential Equations, Acta Mathematica Scientia, 5(2003):613-618. 9. Z.X.Chen, C. C. Yang, Some Further Results on the Zeros and Growths of Entire Solutions of Second Order Linear Differential Equations [J], Kodai Math. J. 22(1999): 273-285.
S H A R I N G VALUES A N D N O R M A L F A M I L I E S
J I A N G - T A O LI Department of Mathematics Chongqing University Chongqing, 400044 P- R- China and Department of Mathematics Sandong University Jinan, Sandong 250100 P. R. China E-mail: [email protected]
Let T be a family of holomorphic functions in a domain D, let k be a positive integer, and let a(z), b(z)(^ 0) and c(z)(jt 0) be holomorphic functions in D. If, for each / e T and a complex number d, all zeros of / — d have multiplicity at least k, L[f] = a whenever/ = 0, / = c whenever L[f] = b, where L[f] denotes linear differential polynomial in / with coefficients holomorphic in D, then T is normal in D. This result extends and improves some normality criterions due to Miranda, Chuang and other authors. Some examples are provided to show that our results is sharp.
Keywords: Holomorphic function, normal families, shared values. 2000 Mathematics
Subject Classification:
30D35, 30D45
1. I n t r o d u c t i o n a n d results We denote by C the whole complex plane. Let / and g be two meromorphic functions in a domain B c C , and let a and b be complex numbers. If g{z) = b when f(z) = a, we write f(z) = a =>• g(z) = b. If f(z) = a => g(z) = a and g(z) = a =>• g(z) = a, we say that / and g share a in D. If / and g share a in C, then we say that / and g share a. Let h be a meromorphic function in C. h is called a normal function if there exists a positive M such that h#(z) < M for all z e C, where [
'
i + IM*)l2
denotes the spherical derivative of h. 137
138
Let T be a family of meromorphic functions in a domain D CC We say that T is normal in D if every sequence {/„} C T contains a subsequence which converges spherically uniformly on compact subsets of D, see [9] or [17]. In 1912, Montel [13] proved Theorem A (Fundamental Normality Test). Let J7 be a family of holomorphic functions in a domain D. If, for each f £ T, f =^ 0, and / ^ 1, then T is normal in D. In 1935, Miranda [12] proved Theorem B. Let J- be a family of holomorphic functions in a domain D, and let k be a positive integer. If, for each f G T, f ^ 0, and f^ ^ 1, then T is normal in D. Chuang [7] (cf. [17, Theorem 4.4.14]) generalized Theorem B as following Theorem C. Let J7 be a family of holomorphic functions in a domain D. Set k
J'=l
where k be a positive integer and ai(z) (i = 0,1, • • • , k — 1) be holomorphic functions in D. If, for each f £ T, f ^ 0, and L[f] ^ 1, then T is normal in D. In this paper, we prove the following result which is further generalization of the above theorems. Theorem 1. Let T be a family of holomorphic functions in a domain D and L[f] be a differential polynomial of the form (1.1). Let a(z), b(z)(^ 0) and C(Z)(T£ 0) be holomorphic functions in D. If, for each f G T and a complex number d, all zeros of f — d have multiplicity at least k, f — 0 => L[f] = a, and L[f] = b => f = c, then T is normal in D.
139
Remark 1. When {z € D : /(z) = 0} = 0 , {z e D : L[/](z) - 6(z) = 0} = 0 , Theorem 1 specializes to Chuang's results, for d = 0. Example 1. Theorem 1 does not hold if the requirement that J7 is & family of holomorphic functions is replaced by that T is a family of meromorphic functions. Indeed, Let n(nz — 1) and D — {z : \z\ < 1}. Then for every / E T, we have ;
_ n + (nz- l ) 2 ~ n{nz - 1) '
;
, _ (nz - l ) 2 - n _ (nz - l ) 2
Thus / = 0 => f = 2, and f'{z) ^ 1 (and hence / ' = 1 =* f = c for any c). But T is not normal in D. Example 2. The following example shows that b ^ 0 is necessary in Theorem 1. Let a be a complex number and D = {z : \z\ < 1 } . Set F={en*-
- : n = l,2,3,---} n Then ^ is a family of holomorphic functions on D. For every / S T, we have f = 0 => f = a, f'(z) ^ 0 (and hence / ' = 0 => / = c for any c). But T is not normal in D. Example 3. The following example shows that c ^ 0 is necessary in Theorem 1. Let T = {^(enz
+ e~nz - 2) : n = 1,2,3, • • • }
and .D = {z : | , z | < l } . Then ^ is a family of holomorphic functions on D. For every / € .7-", we have n / = -L(e * + e-n,!-2), J
/" = enz + e-n2
n Obviously, all zeros of / have multiplicity two, / = 0 =*> / " = 2, / " = 2 => / = 0. But .F is not normal in D. Example 4. The restriction in Theorem 1 on the zeros having multiplicity at least k is necessary. In fact, let T = {n(ez - e"z) : n = 1,2,3, •••}
140
and D = {z : \z\ < 1}, where w ^ 1, u>k = 1, k > 2. Then F is a family of holomorphic functions on D. Obviously, for each / € J", f^ = / , so / and /(fc) share any complex number in D. But F is not normal in D. Xu [19] and Pang [15] proved Theorem D. Let J7 be a family of holomorphic functions in a domain D, and let a and b be two distinct finite complex numbers. If for each f £ F, f and f share a and b in D, then F is normal in D. Chen and Fang [4] extended Theorem D as following. Theorem E. Let Fbe a family of holomorphic functions in a domain D, let k > 2 be a positive integer, and let a, b(^ a) and c be three finite complex numbers. If, for each f £ F, all zeros of f(z) — c have multiplicity at least k, and f and f^ share a and b in D, then F is normal in D. Recently, Fang and Xu [8] improved Theorem D by proving the the following result. Theorem F. Let F be a family of holomorphic functions in a domain D, and let a and b(^= a) be two finite complex numbers. If, for each f G J-', f and f share a in D, and f = b=> f = b, then T is normal in D. Theorem G. Let !F be a family of holomorphic functions in a domain D, and let a and b{^ a, 0) be two finite complex numbers. If, for each f G IF, f and / ' share a in D, and f = b => / = b, then T is normal in D. In this paper, we prove Theorem 2. Let F be a family of holomorphic functions in a domain D and L[f] be a differential polynomial of the form (1.1). Let a he a complex number, and b(z)(^ L[a],a) be a holomorphic functions in a domain D. If, for each f 6 F and a complex number d, all zeros of f(z) — d have multiplicity at least k, f = a => L[f] = a, and L[f] =b=$- f = b, then F is normal in D.
141
As a consequence, we obtain the following result. Corollary 1. Let T be a family of holomorphic functions in a domain D and L[f] be a differential polynomial of the form (1.1) with cto ^ — 1. Let a and b be two complex numbers such that b^ a. If, for each f £ J- and a complex number d, all zeros of f(z) — d have multiplicity at least k, and f and L[f] share a and b in D, then T is normal in D. Proof. Since ao ^ — 1, a ^ b, we deduce that b ^ L[a] or a ^ L[b]. Noting that / = a o L[f] — a, and L[f] —b<^-f = b. By Theorem 2, T is normal in D. • Remark 2. Obviously, Corollary 1 specializes to Theorem D-E, if we take ai = 0 (i = 0, l,--- ,k-l) in L[f\. Let k — 1. From Theorem 2 we immediately obtain the following result which extends and improves theorem F-G. Corollary 2. Let T be a family of holomorphic functions in a domain D, and let a and b be two finite complex numbers such that b ^ aao, a, where ao be a holomorphic function in D. If, for each f£jr,f = a=>f' + ao(z)f = a, and f + ao(z)f = b => / = b, then T is normal in D. It is assumed that the reader is familiar with the basic results and notations of Nevanlinna's value distribution theory, as found in [11]. In particular, we denote by S(r, f) any function satisfying S(r, f) = o{T(r, / ) } as r —» oo, possibly outside a set of r of finite linear measure. 2. Some lemmas Lemma 1([16]). Let T be a family of holomorphic functions in unit disc A with the property that for each f £ T, all zeros of f are of multiplicity at least k. Suppose that there exists a number A > 1 such that \f^(z)\ < A whenever f € T and / = 0. If J7 is not normal in A, then for 0 < a < k, there exist (1) (2) (3) (4)
a a a a
number r € (0,1); sequence of complex numbers zn, \zn\ < r; sequence of functions fn G T; and sequence of positive numbers pn —> 0
142
such that 0 such that
for all z G C\{0} which are not poles of g. Lemma 5(Milloux inequality, cf. [11])- Let g be a transcendental meromorphic function and k be a positive integer, then for b G C\{0}, we have T(r, g) < N(r, g) + N(r, ±) + N(r, -j^—^
- N(r, ^ y ) + S(r, g).
Lemma 6 ([9]). Let T be a family of holomorphic functions in a domain D. If there exists a positive number K such that \f{z)\ > K for all f € T and z £ D, then T is normal in D. 3. Proof of Theorem 1 Proof. It is sufficient to show that T is normal at each point in D. Let z' 6 D. Then there exists a disc A contained (with its closure) in D such that z' £ A and b(z)c(z) ^ 0 for all z G A. Without loss of generality, we
143
assume that A = {z : \z\ < 1}. Suppose that F is not normal in A. We distinguish two cases. Case 1. Suppose that d = 0. Since a(z) is a holomorphic function in A, there exists a finite number M > 0 such that M = sup{|a(z)| : \z\ < 1}. By the assumption we know that | / ^ ( . z ) | = |£[/](z)| = \a(z)\ < M when f(z) = 0. By Lemma 1 (with a — k and A = M + 1), there exist / „ G T, zn —> ^o («n> 2o G A), and pn —> 0 + such that ffn(0=p;;fc/n(*n
+ P n O - S ( 0
(2)
locally uniformly with respect to the spherical metric on C, where g{£) is a nonconstant entire function satisfying that each zero of g has multiplicity at least k, and that g*(O
= k(M + i) + i
(3)
Thus • g(fc) = a(zo) on C Suppose that there exists a point £o such that g(£o) = 0. Then by Hurwitz Theorem, there exist £ n , £ n —> £o as n —> oo such that (for n sufficiently large) 9n(Zn)
= Pnkfn(Zn
+ Pn$n)
= 0
(4)
Noting that pn > 0, / = 0 => L[f] = a for all / G T and that all zeros of f(z) have multiplicity at least k, from (3.1), (3.3) we have 9{nk)(Zn)
=
tik){Zn
+ Pntn)
= L[fn}(zn
+ pn£n)
= a(zn
+
Pn^n)
(5)
Thus oo
This shows that g = 0 => g^ = a(zo). Next we prove that g^(^) ^ b(z0) on C. Suppose that there exists a point £0 such that g^(£o) = b(z0). If #(fc)(£) = b(zo), then g is a polynomial of degree k. Since each zero of g has multiplicity at least k, g must has a single zero £1 with multiplicity k, this implies g(£) = - £ p ( £ — £i)fc, which together with the fact that g = 0 =>• g^ = a(zo) gives that 6(20) = a(zo). By simple computing we have 9
{U>
~ \\a(zo)\,
if | f r | < l
144
so that g*(0) < fc(|a(z0)| + 1) + 1 < k(M + 1) + 1, which contradicts (3.2). Thus ff(fc)(C) ^ b{zQ). Prom (1.1) and (3.1) we have
L[fn}(Zn
+ Pn0
= f™ {Zn + PnO + J2 ak.j j= l
(zn + PnOf^H^n
+
PnO
fc
(6)
Since fc j= l
as n —> oo, thus we have lim {L[/„](z„ + />„£) - b(zn + Put)} = lim {fl^fc)(0 - K*n + PnO} n—KX>
n—*oo
= $<*> ( O - ^ o ) Noting that £o as n —> co such that (for n sufficiently large) L[fn}(zn
+ Pntn)
= Hzn
+ Pn€n)-
(7)
Since L[f] = 6 =>• / = c for all / € .F, from (3.1), (3.6) we have 5 n ( £ n ) = Pnkfn(z„
+ PnCn) = P^
c z
in
+ PnZn)
(8)
But c(zn + Pnin) —» c(zo) as n —> 0 and C(ZQ) / 0, from (3.7) we get g(£o) = lim 5n(£n) = oo which contradicts #'fc'(£o) = b(z0). This shows that ff^(0 7^ &(zo) on C. Since 5 is of order at most one, so is g(k\ it follows that g(k\ti)
= b(z0) + eb°+b^
(9)
where bo, b\ are finite constants. We divide this case into two subcases. Case 1.1. If b\ — 0, from (3.8) we know that g{£) is a polynomial of degree k. Since each zero of g has multiplicity at least k, g must has a single zero £2 with multiplicity k, so that
145
which together with the fact that g = 0 =>• g^ = a(zo) gives thatfr(zo)+ eb° = a(zo). By simple computing we have 9
W
- \
\a(z0)\,
if |&| < 1
so that g#{0) < k(M + 1) + 1, which contradicts (3.2). Case 1.2. If b\ ^ 0, then g is a transcendental entire function with order at most one. Since gW ^ b(zo)(^ 0), by Lemma 5 we deduce that g has infinitely many zeros z\, Z2, •••, and \ZJ\ —> oo as j —> oo. Define /i(z) = gC 5 " 1 )^) - 6(«o)-2, then ft'(z) = 3(fe)(z) - b(z0) + 0, softhas no critical values. Hence, by Lemma 3, h has only finitely many asymptotic values. Applying Lemma 4 toft,we have
WW
^ 1 l o g IM*j)l = 1 l o g Hzo)zi\
\h(zj)\
~ 2?r
b
R
2?r °
R
This implies \zjh'(zj)\
oo (as j —> oo)
(10)
Recalling that all zeros of g(£) are of multiplicity at least k and that g = 0 =£. ^(fc) = a(zo), we have |z,-ft;(zj)| _ \h(Zj)\ '
a(z0)-b(zo) 6(^o) '
^
j
Prom (3.9) and (3.10) we deduce a contradiction. Thus T is normal at z'. Case 2. Suppose that d ^ 0. If k = 1, then the proof is completed as in Case 1. We now assume that k > 2. Applying Lemma 1 to a = 1, we can find / n £ f , z„ -* 2o (^n, ^o S D), and p n —> 0 + such that 9n(0 = fn(Zn+Pn0-d-^g(0
(12)
locally uniformly with respect to the spherical metric on C, where g(£) is a nonconstant entire function satisfying that each zero of g has multiplicity at least k > 2, and that g is of order at most one. We claim that g(£) ^ —d. For otherwise, there exists £o such that
9(0 ^-d
(13)
Then by Hurwitz Theorem, there exist £„, £n —> £o as n —» oo such that (for n sufficiently large) ~d = gn{£n) = fn(zn
+ Pntn) ~ d.
(14)
146
Thus we get fn(zn + p„£ n ) = 0. Since / = 0 =*> L[f] = a for all / € T and all zeros of f(z) have multiplicity at least k, by (3.11) we have 9{nk)(Zn)
= Pnfik\zn
+ PnU)
= ptHfnKZn
+ PnZn) = p\a{zn
+ />„,£„)
This implies 2(fc)(£o) = lim < # > « „ ) = ( ) .
(15)
n—>oo
If g(k\£) = 0, then g is a polynomial of degree < k, and so could not have zeros of multiplicity at least k, a contradiction. Thus s ^ ( £ ) ^ 0. From (1.1) and (3.11) we have lim p„{L[f okJ n](zn n—>oo
+ pn0 - b(zn + /?„£)}
pkL[fn](zn+pn£)
lim
k =
k
l i m {p J^\zn
+
Pn0
k
+P nY^ Ok-jiZn 3=1
+ PnOfLk-J)(Zn
+
PnO}
fc-1
= lim {<#> (£) + J2 P>nak-i (zn + pn09(k-j)
(0
i=i
+ p£a 0 (z„ + /?„0[<7n(0+d]} = lim 5 i fc) (^) n—»oo
fl(fc)(0
=
(16)
By Hurwitz Theorem, from (3.14) and (3.15), we know that there exists £„, £« —> Co as n —> co such that (for n sufficiently large) pkn{L[fn}{zn + Pn(n) - b(Zn + pn(n)} which leads to L[fn](zn + pn£,n) = b(zn + pn£n)that L[f] = b =>• / = c for all / £ T, we have 9n(£n)
= fn(zn
+ Pn£n) - d = c(zn
=0
(17)
From this and recalling
+ pn£n)
- d
(18)
It follows that g(£o) = lim gn(£n) = c(z0) - d
(19)
n—>oo
Combine (3.12) and (3.18) we get c(zo) = 0, a contradiction. This shows that
147
zeros and only simple zeros. This contradicts the fact t h a t all zeros of g have multiplicity at least k > 2. T h u s T is normal at z' e D. T h e proof of Theorem 1 is completed. • 4. P r o o f o f T h e o r e m 2 P r o o f . Set f = { / : / = / - a; f € T). Then zeros of / — (d — a) have multiplicity > k, / = 0 => L[f] = ^ - •f'(a) => f = b — a. Since 6 - L[o] ^ 0, Theorem 1 t o F, we know t h a t T is normal in D. normal in D. T h e proof of Theorem 2 is completed.
for each / € F, all L[/] = a - L[a], and b - a =£ 0, applying It follows t h a t J7 is •
References 1. W. Bergweiler, On the zeros of certain homogeneous differential polynomials, Arch. Math. 64 (1995):199-202. 2. W. Bergweiler, Normality and exceptional values of derivatives, Proc. Amer. Math. Soc. 129 (2001):121-129. 3. W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995):355373. 4. H. H. Chen and M. L. Fang, Shared values and normal families of meromorphic functions, J. Math. Anal. Appl. 260 (2001):124-132. 5. H. H. Chen and Y. X. Gu, An improvement of Marty's criterion and its applications, Sci. China Ser. A 3
148
16. X. C. Pang and L. Zalcman, Normal families and shared values, Bull. London Math. Soc. 32(2000):325-331. 17. J. Schiff, Normal Families, Springer-verlag, Berlin, 1993. 18. W. Schwick, Sharing values and normality, Arch. Math. 59(1992):50-54. 19. Y. Xu, Normality criteria concerning sharing values, Indian J. Pure Appl. Math. 30 (1999):287-293. 20. L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc. 35(1998):215-230.
T H E BLOCH T Y P E SPACES VIA T H E CESARO MEANS*
J I N G LIN Department Shantou,
of Mathematics, Shantou Guangdong, 515063 P. R. E-mail: [email protected]
University China.
Using the Cesaro means of Taylor extensions of holomorphic functions / on the unit disc, an equivalent condition for / to belong to the Bloch type spaces is given. 2000 Mathematics
1.
Subject Classification:
30H05 and 46E15
Introduction
Throughout this paper we denote if (D) the set of holomorphic functions oo
on the unit disc D. For f(z) = ^ anzn € if (ID) the Cesaro means of the n=0
power series of / is defined by
fc=o
n + i
n +
fc=o
i
where n
Sn{f){z) = Y^akZk,
n=
0,1,2,...,
fc=0
is the partial sums of / . Holland and Walsh [1] gave criteria for a function / in if (D) to belong to the Bloch space B in terms of the Cesaro means of the Taylor series of / . In this paper we consider the Bloch type spaces by studying the Cesaro means. For 0 < a < oo, a function / in if (D) is said to belong to the a-Bloch space, denoted by / £ Ba, if
n/iiBQ = sup(i-i*i 2 n/'(*)i
"This work is supported by NNSF of China (No. 10371069) and the NSF of Guangdong (No. 04011000). 149
150
Note that 1-Bloch space B1 is just the classical Bloch space. Denote by H°° the space of all functions in if (ID) with H/lloo =SUp|/(z)| < 00. 26D
Our main result in this note is the following theorem. 3 Theorem Let / e H (D) and 0 < a < §. The following statements are 2equivalent: / e Ba;
(i) (ii)
\\{n + m + l)*n+m(f)
- (n +
= 0{y/nm{n^~^m^~^
+n
1)
Q_1
+ m
- (m + l)
Q_1
))
(n,m -+ 00);
1
(iii)
||<72n(/)-(rn(/)||00 = 0(n"- ). If a=l, the conclusion is Theorem 2 in [1].
2. Preliminaries To prove our main theorem we consider the Hadamard product of two 00
00
functions f(z) = Y2 anZn and g(z) = J2 bnzn in H(3) as the following n=0
n=0 00
f*9(z)
= ^T,ar>.bnZnra=0
Using the Hadamard product we know that the Cesaro means of / € H(B) can be written as M / ) = °n{k) * / , 00
where k{z) = 1/(1 - z) = £ z " , z e D . Also, we have 71=0 f-2
*n(/) = ^ f " In Jo
Kn{6)f{ze-ie)d6,
where Kn{tf)-^(l-—)e
-
{ n +
1){1_cose)
k=—n
is Fejer's kernel. Note that Kn(8) > 0 and cr n (l) = 1 for n = 0,1, ... .
151
Let Aa, 0 < a < 2, denote the space of functions / in H(V>) for which IIMIUo, = |fl(0)| + \\g\\Aa < o o , where
Iff
\\9\\Aa
7T Jo
1
r2ir
rewg\rew)
+ g{rew) -
(2.1)
JO
Note that Aa contains constant functions only for a > 2. The following result will be used in the proof of our main theorem and it is also an extension of Theorem 2.3 of [2]. oo
L e m m a 1 Suppose 0 < a < 2. Let f = Yl anZn £ Ba and g{z) = n=0 n
£ bnz
£ Aa. Then
n=0
l/*^)l
(2.2)
Proof. For z = re%e € D and ^ € ED, it is easy to see that £ arAf1"1 = - / / n^l ^ Jo Jo
(1 - r2)f'm
- \(zg(z) dZ
g(0))]e-ied9dr.
Since / S B a , we have oo
<
Iff
1
/-27T
\rewg'{rew)
+ g(rew) - 5 (0)|(1 -
rIy-ad6dr,
* Jo Jo and so
^2 a"bnC
n=0
which completes the proof.
D
The following lemma will also be used in the proof of our main result (see [5]). L e m m a 2 (Bernstein Theorem): Suppose T(x) is a real polynomial of order n, and M = max|T(a:)|. Then \T'{x)\ < 2nM.
152
3. The Proof of the Main Theorem Prom now on, we begin to prove the main theorem.
Proof. (i)=^(ii). Ourproofwillbegivenby Lemma 2. Noting that
- (n +
1)
- (m + l)am(f)
an(f)
= gn,m * / ,
where 9n,m - 9m,n = (n + TU + l)(Tn+m(k)
- (n + l)o"n(fc) - (m +
l)(Tm(k).
Now for n = 0,1,2,..., On{k)(z) = £ ( 1 - ^ V , z(l-zn+1) (n+l)(l-z)2
n = 0,1,2,..., 1 1-2
Hence _ z ( l - z " + 1 ) ( l - zTO+1) <7n,m(z) — Q _ £\2
To establish the assertion, it suffices, in view of Lemma 1, to show that \rei6g\reie)
hn,m\\Aa = - / / n Jo J0
= 0{ypmn{n%~^m^~*
+ g(reie) - g(0)\(l + na_1
r2f-adrdB
+ma~1))
a s n , m - » oo. With this in mind, we observe that, if n , m = 1,2,..., then z
9n-l,m-l(z)
=
+
9n-l,m-l(z)
-j-Z9n-l,m-l(z)
_ 2z(l - zn)(l - zm) ~ (l-z)3
nzn+1(l-zm) (I-*)2
~
mzm+1(l-zn) (l-^)2 '
According to the definition of the norm of Aa, we have rl
\\9n,m\\Aa ~
2z(l - zn)(l - zm) (1-z)* n Jo Jo mzm+1{l-zn) (l-r2y-adrd9 (l-z)2
—
r2n
/
Iff
1
r2
^z{l-zn)(l-zm)\ (1 - r'y-adrd6 \l-z\
*
Trio Jo +
nzn+1(l-zm) [l-zf
/
nj0
J0
|(1-Z)2|
^
+
dTd
">
^
^I'jy^y-^^-r^drdO |(l-z)*| = h(n,m)
+ I2(n,m)
Now we estimate Ii(n,m), To deal with I\(n,m),
l2(n,m)
and I^n^m),
respectively.
the Holder inequality shows that
7T y 0 Jo 1
<
+ /3(n,m).
f 27T
Wo io
<
|i - -z|3 |(l-zn)(l-zm)| (1 \l-z\-
a
ry-
rdrd6
(h^Hhim))^,
where rL rZ7r h _
hM=
/•l
r2ir | i _
/ ./o Jo
Now we estimate I\(n).
\
W p ( 1 - r ) '**•
LL
h{m)=
n 2
z
m|2
'M1 — z .I ( l - r 2 ) 1 - " ^ ^ . I-l
If A is not zero or a negative, then (also see •,
i
(\-zwY
+oo
= *-> V ^^W. n\T(X)
Letting A = | , we obtain
(1 - 2 ) 3 / 2
^ C ^ j=0
= ^7(i)^. i=0
154
Define c s = 0 if s < 0, and we have (1 -zn)n)
~*
== f V - c - )z«
3
»=0 Z ^6*
(1 - z) /2
C
*-«-lZ '
Hence rl °°
/i(n) = 4 / •^
V | Ci -
Ci_n|
2 2i+1
r
(l - r2)1_Qdr
i=o
oo
/-l
= 4^|Ci-Ci_„|2/ <=o
r^+Hl-r2)1-^
^
OO
= 2^|Ci-Ci_n|2/(i+l)2-Q i=0
ra-l
= 2^
| C i | 2 /(i + I ) 2 " " + 2 ^
| Ci _„ -
Ci|
2
/(i + 1) 2 ~"
i=n
i=0
= 2(Ji + J 2 ). We know that Ci = 7(»).
« = 0,1,2,....
For t > 0, let r ( t + 3/2) r(3/2)r(t + i ) -
7(*)
fur By the properties of the T function (also see [1]), it is easy to show that (t + 3/2)
7'(*)=7(*){YJ r ( t + 3/2) ,
r ' ( t + l) r ( t + i)
^ 1/2 ^ ( n + t)(n + 4 + 1/2) = 7(«)EN
t=
1/2 <*)£ t=i (« + * - l / 2 ) ( n + t + l/2) OO
7(t)/(l
+ 2t).
By the above inequality, we can show that ry(t)/y/2t+ [0, oo). Therefore, for t > 0, 7(t) < V 2 t + 1 and 7#(t) <
l/y/2t+T.
1 is decreasing on
155
Particularly, 7(2) = a < y/2(l + i). Thus "-1
h'2
*=Erfib=w Now we begin to handle J2. Note that if i > n, then 0 < Ci - Ci_„ = /
7'(t)dt
^
\ / ( l + 2*)
JK
= v/(l + 2t) - 7 ( 1 + 2i<
2n)
2n \/rT2i'
Therefore J 2 = 5 3 l c j _ „ - c i l 2 / ( t + l)
<
2-Q
—
£ (l + 2i)(i4rr+ l ) 2
z=n
/>oo
= 4n 2 / (a; + Jn =
l)a-3dx
0(na).
Hence h(n) = 2(J1 + J2) =
0(na),
and then by the Cauchy-Schwarz inequality, we obtain that
I\{n,m) =
0(n^m^).
156
Next we begin to estimate /2(n,m). Indeed, we can write l2{n,m) as 1 /-27T
h(n,m) = ^ [ r * Jo Jo
\^\{i-r2)'-rn+ldrdB
|l-z|
2f n
Wo
70
_,
r /-27T ii
/-l
Wo /
m JmBi2 9
\7O fl
|1
_rmeim9i dOdr a,n
|l-re« |2
/^-n-
70
-.
|l-re^|
~> 1/2 2
J
j
rn+1(l-r)2_Qrfr
JO
3 = Cny/mB(n+2, - - a )
Similarly we can obtain that I3(n,m)
= 0{ma~%n%).
We can summarize that ||ffn,m|Ua = 0 ( y W i ( n 2-
-
1
- -
2m2
1
,Q-1 i ™ a - l \ 2 + „<* + m « i)).
(ii)=>(iii). Assume that (ii) holds and take m = n in (ii). The triangle inequality gives (2n+l)||<7 2 n (/)-<7 n (/)||oo< To estimate
||O"TI(/)||OOJ
we
K(/)||oo
+0(na).
obtain by letting m = 1 in (ii),
(n + 2 ) | | a „ + 1 ( / ) | | 0 0 < ( n + l ) | | < T „ ( / ) | | 0 0 + ||a 1 (/)|| 0 0 + 0 ( n t + n Q - 5 ) < ( n + l)||<7n(/)||oo + 0 ( n * + n a - * ) . That is to say lk„ + i(/)||oo <
j^hn{f)\\oo+0{n^+na-3*)
^IW/Jlloo + O ^ ^ + n 0 - ? )
+na~^).
157
Thus, ( 2 n + l ) | k 2 n ( / ) -
=
0(na).
Hence ||
(n -> oo).
Taking n = V and summing on j from j — 0 to j = m, we obtain K " ( / ) l l o o < ||CT^ m (/) - 0"i
x(/)||oo + ||<^™-i(/) - ^ 2 ™ - » ( / ) l l ~
+ - - - + K . ( / ) - ^ ( / ) H o o + ||^o(/)||oo
= (9((2 m - 1 ) a ) + 0((2 m - 2 ) Q ) + ... + 0((2°) Q ) + 1 = (9((2m)Q). Let Vm = (2 + 2 - " > 2 m + 1 (fc) - (1 + 2-m)a2m(&). Then Vm*f
= (2 + 2'm)a2m+1 (/) - (1 + 2 - " > 2 m (/).
Hence \\(Vm * /)'||oo = 11(2 + 2-m)a2m+1(f)
- (1 + 2 " " > 2 m ( / ) | | 0 0 m
= K m + l ( / ) + (1 + 2" )( ( T 2 m + 1 (/) - ^ m ( / ) | | o o < lk 2m+1 (/)lloo + (1 + 2 — ) | | ( a 2 m + 1 ( / ) - ^ ( f l l l o o = 0((2 m ) Q ) + (1 + 2 - m ) 0 ( ( 2 m ) Q ) = 0((2 m ) Q ). Note that o-n(Mo/') = Mo<^(/) holds for the identity function /i 0 on D and then Hcr^/i^loo < \\h\loo whenever h G if 00 . Furthermore, the first 2 m Taylor coefficients of / and Vm * f are equal.
158
We put these three facts together to prove t h a t ||0^(/)||oo = 0(na) n —> oo. So , let n > 1 and choose m so t h a t 2 m _ 1 < n < 2m. T h e n \\H><(f)\\oo
as
= lkn(/io/')lloo = \\
oo
For / (2) = 5Z an-z™ and z = r^, it is easy to obtain n=0 oo
zf'(z)
= (1 - r) 2 ^ ( „ + l ) ^ ( / ) ( O r " . n=0
Furthermore oo
r | / ' ( z ) | < (1 - r) 2 J >
+ l)K(/)(0|r"
n=0 oo
< C ( l - r ) 2 ] T ( n + l)narn n=0
^ (1 - r)a ' T h a t is to say f € Ba. We have completed the proof of (iii)=>(i).
D
References 1. F. Holland and D. Walsh, Criteria for membership of Bloch spaces and its subspaces BMOA, Math. Ann, 273 (1986), 317-335. 2. J. M. Anderson, Clunie and Ch.Pommerenke, On Bloch functions and normal functions, J.Reine Angew. Math, 270, (1974), 12-37. 3. H. Wulan and J. Zhou, QK type spaces of analytic functions. Journal of Function Spaces and Applications, 4(2006), 73-84. 4. E. Titchmarsh, Theory of functions, 2 n d ed, Oxford: University Press 1939. 5. A. Zygmund, Trigonometric Series, I, II. Cambridge: University Press 1968. 6. T. Korner, Fourier Analysis, Cambridge University Press, London , 1988. 7. K. Zhu, Operator theory in function spaces, New York, Marcel Dekker, 1990.
UNIQUENESS OF MEROMORPHIC FUNCTIONS CONCERNING WEAKLY WEIGHTED-SHARING
SHANHUA LIN Department of Mathematics Fujian Normal University Fuzhou, 350007 P. R. China E-mail: [email protected] WEICHUAN LIN Department of Mathematics Fujian Normal University Fuzhou 350007, P. R. China and Department of Mathematics Shantou University Shantou 515063, P. R. China E-mail: [email protected] In this paper, we introduce the definition of weakly weighted-sharing which is between "CM" and "IM". Using the notion of weakly weighted-sharing, we study the uniqueness problems on meromorphic function and its fcth order derivative / ' * ' . As consequences, we are able to answer questions posed by Kit-wing Yu, which were also studied by L. P. Liu and Y. X. Gu. Our results sharpen the above results.
Keywords: Weakly weighted-sharing, uniqueness, meromorphic function. 2000 Mathematics Subject Classification: 30D35, 30D45
1. I n t r o d u c t i o n a n d M a i n R e s u l t s In this paper, we assume t h a t the reader is familiar with the basic concepts of Nevanlinna value distribution theory and the notations, see e.g. [1, 2]. We denote S(f) the set of all small functions of / . For any two nonconstant meromorphic functions / and g, and a G S(f)n 159
160
S(g). Let Ng(r, a) be the counting function of all common zeros of / — a and g - a with the same multiplicities, and N0(r, a) be the counting functions of all common zeros of / — a and g — a ignoring multiplicities. Denote by NB(r, a) and N0(r, a) the reduced counting functions of / and g corresponding to the counting functions NB(r, a) and N0(r, a), respectively. If N(r, S(r, g)(N(r,
— — ) + N(r, - — ) - 2NB(r, j a g a
7 ^ - ) + ^ , j &
— ) g a
a) = S(r, f) +
- 2JV0(r, a) = S(r, / ) + S(r,
then we say that / and g share a "CM"("IM"). When N(r,
g)),
-^—)
+
N(r,
) = 2NB(r, a)(N{r, —!—) +N(r, -?—) = 2N0(r, a)), then g- a J - a g -a we say that / and g share a CM(IM). In 2003, Kit-wing Yu[3] proved the following results. Theorem A. Let k>l. and a =£ 0,oo. / / / , f^
Let f be a non-constant entire function, a € S(f) share a CM and 6(0, / ) > | , then f = /( fc ).
Theorem B. Let k > 1. Let f be a non-constant non-entire meromorphic function, a £ S(f) and a ^ 0,00, / and a do not have any common pole. If f, f{k) share a CM and 46(0, / ) + 2(8 + fc)9(oo, / ) > 19 + 2k, then In the same paper, Kit-wing Yu posed the following open questions: Question 1. Can a CM shared value be replaced by an IM shared value in Theorem A? Question 2.
Is the condition 6(0, f) > 4 sharp in Theorem A?
Question 3. Is the condition 46(0, / ) + 2(8 + A;)9(oo, / ) > 19 + 2k sharp in Theorem B? Question 4. Can the condition " / and a do not have any common pole" be deleted in Theorem B? In 2004, L. P. Liu and Y. X. Gu[4] obtained the following results which answered questions 2 and 3. Theorem C. Let f be a non-constant meromorphic function, a £ S(f) and a ^ 0,00. / / / , f^ share a CM, f^ and a do not have any common pole of same multiplicity and 26(0, f) + 40(oo, / ) > 5, then f = f^k\
161
Theorem D. Let f be a non-constant entire function, a G S(f) and a ^ 0,oo. Iff, /(*> share a CM and 6(0, f) > \, then f = /<*). In this paper, we introduce the definition of weakly weighted-sharing. By the new definition, we obtain uniqueness theorems which answer the questions posed by Kit-wing Yu. Moreover, our results improve Theorem A, B, C, D mentioned above. Next, we introduce some notations for our definition. Definition 1. Let f and g be two nonconstant meromorphic functions sharing "IM", for a G S(f) D S(g), and a positive integer k or oo. (i) Nk^(r, a) denotes the counting function of those common a-points of f and g with the same multiplicities, both of their multiplicities are not greater than k, where each a-point is counted only once. (ii) N ,k(r, a) denotes the counting function of those common a-points of f and g, both of their multiplicities are not less than k, where each a-point is counted only once. Definition 2.
For a G S(f) n S(g), if k is a positive integer or oo, and
Nk)(r, j^-a)-NEk){r,
N(k+i(r,
a) = S(r, / ) ,
Nk)(r,
-^—)-N°{k+1(r,
9
-i-)-N^(r,
a) = S(r, g);
a) = S(r, / ) ,
"
Or if k — 0 and N(r, - J — ) - A T 0 ( r , a) = S(r, / ) , f-a
N(r, -L-)-N0(r, g-a
a) = S(r, g),
then we say f and g weakly share a with weight k. Here, we write f, g share "(a, k)" to mean that f, g weakly share a with weight k. Obviously, if / and g share "(a, k)", then / and g share "(a, p)" for any p(Q
162
T h e o r e m 2. Let k > 1, and let f be a non-constant meromorphic function, a G S(f) and a ^ 0, oo. If / , f^ share a "IM" and 56(0, f) + (2k + 7)0(00, / ) > 2k + 11, then f = /(*>. If / is a nonconstant entire function, then 0(co, / ) = 1. So we have the following results. Corollary 1. Let k > 1 and 2 < m < oo. Let f be a non-constant entire function, a G S(f) and a ^ 0,oo. If f, /(fc) s/iare "(a, m)" and
(5(0, / ) > £ , thenf = f^. Corollary 2. Let fc > 1, and let f be a non-constant entire function, a G 5S(f) ( / ) and a ^ 0,oo. 7 / / , f^ share a "IM" and 6(0, f) > f, then
R e m a r k 1: Theorem 1 and Corollary 1 improve Theorem A~D. Theorem 2 and Corollary 2 answer question 1. Meanwhile, we give an affirmative answer to the fourth question. 2. Some Lemmas L e m m a 1[5]. Let f be a nonconstant meromorphic function and let k be a positive integer. Then (i) N(r> yrfej) < *(r, } ) + kN(r, f) + S(r, f). (n)N(r,
1 ) < T ( r , /<*>)-T(r, / ) + N(r, j) + S(r, / ) .
Next, we introduce some notations for the following lemma. When / and g share 1 "IM", TV" (r, ——-) denotes the counting function of the 1-points of / whose multiplicities are greater than 1-points of g, where each zero is counted only once. Similarly, we have N (r, Nn(r,
-—-)
-).
denotes the counting function of common simple 1-points of
/ and g. L e m m a 2. Let m be a nonnegative integer or oo. Let F and G be two nonconstant meromorphic functions, and F, G share "(1, m)". Let //
IfH^O, then
F"
2
F'
-(^" F^T
)_(
G"
2
G'
G^" G^T)-
163
(i)If2<m<
oo, then
T(r, F) < N2(r, F)+N2(r,
~) + N2(r, G)+N2(r,
^) + S(r, F) + S(r, G).
(ii) If m = 1, then T(r,F)
^) + NL(r,
-^-)
F) + N2(r, ^) + N2(r, G) + N2(r, ±) + 2NL(r,
-^-j)
F) + N2(r, j) + N2(r, G) + N2(r,
+S(r, F) + S(r, G). (iii) lfm = 0, then T(r, F)
+1 L r
* ( > G^l)
+ 5 r
( '
F ) + 5(r
The same inequalities holds for T(r,
'
G)
-
G).
Proof, (i) If 2 < TO < oo, then by the Second Fundamental Theorem, we have T(r, F)
F) + N(r, j) + N(r, j^)-N0(r,
^j) + S(r, F), (2.1)
where No(r, w ) is the counting function of those zeros of F' which are not r
the zeros of F(F — 1). No(r, -^7) can be denned similarly. F" F' By a simple calculation, any pole of F is not a pole of -=^- — 2 , any F F —1 G" G' pole of G is not a pole of — — 2 — — - . Furthermore, let z\ be a common Gr
Gr — 1
zero of F — 1 and G — 1 with multiplicity t, where 1 < t < 2. We know that H is analytic at z\. Therefore, by H ^ 0, we have N1}(r, jL^)
jj)+S(r,
F)+S(r, G) < T(r, H)+S(r,
F)+S(r,
G). (2.2)
Note that m(r, H) = S(r, F) + S(r, G) and N(r, H)
F)+N{2(r,
N0(r, jj) + N0(r, ~)+NL{r,
G) + N{2(r,
j) + N{2(r,
^ J + J V ^ r , ^-[)+S(r,
i)+
F)+S(r,
G).
^)+N0(r,
jj)
By (2.2), we have Ni)(r, p^rj)
F)+N(2(r,
G)+N{2(r,
j)+N(2(r,
164
+ NL(r,
+No(r, ^)
L
yL_)+N
(r,
^ 1 _ ) + S( r , F) + S(r, G). (2.3)
Since F and G share "(1, m)", we have N(2(r, ^rj)+NL(r,
^ ^ N ^ r ,
Jl-J)+N0(r,
±)+N(r,
1)-
N(r, ±) < N(r, ±r). It follows from Lemma 1 that + NL(r,
N(2(r, g^j)
+ NL(r,
JJ-[)
^)+N(r,
^L.)
+ N0(r,
±)
G) + S(r, G).
(2.4)
In addition, we have
Combining (2.1), (2.3), (2.4) and (2.5), we obtain T(r, F) < N2(r, F) + N2(r, ^) + N2(r, G)+N2(r,
±) + S(r, F) + S(r, G).
(ii) If m = 1, then (2.4) is replaced by Ni2(r,
Q^Y)+NL(r,
^-J)+No(r,
±)
±)+N(r,
G)+S(r,
G).
Similar to the arguments in (i), we see that (ii) holds. (iii) If m = 0, then by a simple calculation, any common simple zero of F — 1 and G — 1 is a zero of H. Therefore, by H ^ 0, we have JVn(r, j ^ ) < N(r, jj)+S(r,
F)+S(r, G) < T(r, H)+S(r, F)+S(r,
G).
(2.6) Thus N
"(r>
F
^
+No(r, ^)
^
7
*
+ NL(r,
F
)+X(2(r,
^
^
)
G)+N(2(r,
+ NL(r,
^)+N{2(r,
^)+N0(r,
-^)
^ . J + Sfo F) + S(r, G). (2.7)
165
By the Second Fundamental Theorem, we have T(r, F) + T(r, G) < N(r, F) + N(r, 1 ) + N(r, —!—) - N0(r, - 1 ) F F _ 1 F 1 1 1 +N(r, G)+N(r, -)+N(r, ^—J-Noir, -^)+S(r, F)+S(r, G). (2.8) In addition, we have N
^
^
+ Nir,
^-j)
= 2JV(r, ^ Y ) + 5(r, F) + 5(r, G)
< ^ n ( r , p^)+7fL(r,
p^j)+T(r,
G) + S(r, F) + S(r, G).
Combining (2.6), (2.7) and (2.8), we obtain T(r, F) < N2(r, F) + N2(r, h + N2(r, G) + N2(r, h + 2NL(r,v F' *v' ' ^ ' G' ' +NL(r,
^Lr[)
X
* F-V
+ S(r, F) + S(r, G).
Lemma 3[6]. Let f be a transcendental meromorphic function and a(^ 0,oo) be a meromorphic function such that T(r, a) = S(r, f). Let b and c are any two finite nonzero distinct complex numbers. Ifip = a ( / ) " ( / ^ ) p , where n(> 0), p(> 1) and k(> 1) are integers, then (p + n)T(r, f)<(p -N(r,
+ n)N(r, f)-N(r,
j) + N(r, ^—^ -1) + S(r, V
+ N(r,
-^—)
f).
3. Proofs of Main Theorems Proof of Theorem 1 Let F=J~, G=-—. (3.1) a a Then it is easy to verify F and G share "(1, m)". Let H be defined as in Lemma 2. Suppose that H ^ 0. It follows from Lemma 2 that T(r, G)
166
Using Lemma 1, we have T(r, fik))
f) + N2(r, -f) + N2(r, /(*>) + N2(r, -±)
+ S(r,
f)
/ ( f c ) ) - T ( r , / ) + N(r, -f) + 4N(r, f) + S(r,
j)+T(r,
f)
i.e. T(r, f)<2N(r,
y ) + 4 i V ( r , / ) + S(r, / ) .
It follows that 26(0, f) + 49(oo, / ) < 5, which contradicts 2<5(0, / ) + 49(co, / ) > 5. Therefore H = 0. That is F" F>
F' G" n G' 2——- = — - 2 F-l G' G-l
n
It follows that 1 A F-l G-l + B, where A(y£ 0) and B are constants. Therefore, (B + 1)G + (A-B-1) BG + (A-B) •
[6
'
and T(r, F)=T(r,
G) + S(r, / ) .
Now we distinguish the following two cases. Case 1. Suppose that B ^ —1,0. If A - B - 1 ^ 0, then from (3.2), we have ~N(r,
-A
5
—r) =
By the Second Fundamental Theorem, we have T(r, G) < N(r, G) + N(r, ^) + N(r,
/_ G+
T(r, f{k))
B
_ x) + S(r, G), i.e.
B +l
f) + N(r, -~) f)+T(r,
+ N(r, j) + S(r, / )
/ ( f c ) ) - T ( r , / ) + N(r, -f) + N{r, j) + S(r, / ) ,
and so T(r, f)<2N(r,
j) + N(r, f) + S(r, / ) .
167
It follows that 2d(0, / ) + 6(oo, / ) < 2, which contradicts 2<5(0, / ) + 49(oo, / ) > 5. Therefore, A - B - 1 = 0. From (3.2), we obtain N(r, —LT)
= N(r,
F).
G+ B Similar to the arguments in the above, we also have a contradiction. Case 2. Suppose that B = — 1.
If A + 1 ^ 0. Then from (3.2), we have 7\f(r, - — ^ - — - ) = ~N{r, F). Similar to the arguments in Case 1, we can get a contradiction. Therefore A + 1 = 0, then from (3.2), we have FG = 1. From (3.1), we have ff(k)
= a2.
(3.3)
In the following, we distinguish two subcases. a) If / is a rational function, then a becomes a nonzero constant. So from (3.1), we see that / has no zero and pole. Since / is nonconstant, this is a contradiction. b) If / is transcendental then by Lemma 3, we get in view of (3.1) 2T(r, / ) < 2N(r, I ) + 2T(r, / / « ) + S(r,
f)
< 2N(r, j) + S(r, / ) < 2N(r, ^ ) + S(r, / ) = S(r, / ) , This is a contradiction. Case 3. Suppose that B = 0. If A - 1 + 0, then from (3.2), we have M r , ——— -) = ~N(r, - ) . K K ' G + (A-1)' F' Similar to the arguments in case 1, we also have a contradiction. Therefore . 4 - 1 = 0. From (3.2), we have F = G, this implies / = f^. This completes the proof of the Theorem 1. Proof of Theorem 2. Using Lemma 2(iii), note that XL(r,
^ - y ) < N(r, ^ ) < N(r, g ) + S(r,
f)
G) + N(r, i ) + S(r,
f)
f) + N(r, - ^ ) + 5(r, / ) .
168
and
F) + N(r,
j ) + S(r,
f) + N(r,
j) + S(r,
f) /).
Similar to the arguments in Theorem 1, we see t h a t Theorem 2 holds. References 1. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. 2. L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993. 3. Kit-wing Yu, On entire and meromorphic functions that share small functions with their derivatives, J. Inequal. Pure and Appl. Math., 4(l)(2003):l-7. 4. L. P. Liu and Y. X. Gu, Uniqueness of meromorphic functions that share one small function with their derivatives, Kodai. Math. J. , 27(2004):272-279. 5. H. X. Yi and C. C. Yang, Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, 1995(In Chinese). 6. I. Lahiri and S. Dewan, Inequealities arising out of the value distribution of a differential monomial, / . Inequal. Pure Appl. Math., 4(2)(2003): Art.27.
UNIQUENESS THEORY OF M E R O M O R P H I C FUNCTIONS IN AN ANGULAR DOMAIN
WEICHUAN LIN Department of Mathematics Shantou University Shantou, 515063 P. R. China and Department of Mathematics Fujian Normal University Fuzhou, 350007 P. R. China E-mail: [email protected] SEIKI MORI Department of Mathematical Sciences, Faculty of Science Yamagata University Yamagata, 990-8560, Japan E-mail: [email protected]
In this paper, we deal with a problem of uniqueness for meromorphic functions defined in the whole plane C under certain value/set-sharing conditions in a sector instead of the plane.
Keywords: Shared-set, uniqueness, meromorphic function, angular domain. 2000 Mathematics Subject Classification: 30D35
1. I n t r o d u c t i o n a n d M a i n R e s u l t s In this paper, for a discrete subset 5 of C U{°°} w e P u t E(S, f) •= \JaeS E(a, f) with E(a, f) := {z e C\fa(z) = 0}, where fa(z) = f(z) - a when a € C and foo(z) = l/f(z), and all the roots are counted according to its multiplicity (CM). Given a domain X C C, a set S C C U{°°} a n d 169
a
meromorphic function
170
/ in C, we define a 'set' EX(S, f) C C similarly to the above when X = C:
Ex(SJ)=
\J{zeX\fa(z)
= 0, CM},
a£S
where X is the closure of the domain X in C. Let / and g be two nonconstant meromorphic functions. If Ex{S,f) = Ex(S,g), we say / and g share the set S CM (counting multiplicity) in X. When S = {a}, we also say / and g share a CM in X. We assume that the reader is familiar with the basic results and notations of Nevanlinna's value distribution theory (see [1, 2]), such as T(r,f), N(r,f) and m(r,f). Meanwhile, the order p and hyper-order a2 of / are defined by P •= P(f) = limsup r _oo
p^ log r
-,
log log T(r,f) CT2 •= o-2(/) = limsup ——. log r r_*oo Here, we denote M(<j2){E(o2]) by the set of transcendental meromorphic (entire) functions of finite hyper-order. Since R. Nevanlinna proved his celebrated 'four-CM' and 'five-IM' theorems, there have been given a lot of results on the uniqueness of meromorphic functions in the plane C under a variety of value-sharing conditions. Although examples such as e z and e~z with values 0,1 and oo show that 'four' and 'five' in those theorems cannot be replaced by any smaller number, the Weierstrass Product Theorem says that 'three-CM' is in general a sufficient condition to deduce the uniqueness of meromorphic functions in C with no exceptional values. In connection to this fact, F. Gross (see [3, 4]) proposed an idea of setsharing conditions and asked about the minimum number or the smallest cardinality of the sets which imply the uniqueness of meromorphic functions sharing those sets. Let Sj(j = 1,2,3) as follows: Si = {0}, S2 = {oo} and S3 = {w\wn(w + a) —b = 0}, where n £ N, and the algebraic equation wn(w + a) — b = 0 has no multiple roots. In 1998, H. X. Yi (see [5]) answer the question posed by Gross (see [4]) and proved the following: Theorem A. Let n £ J V \ { l } . If/ and g are two entire functions satisfying Ec{Sj,f) = Ec{Sj,g) for j = 1,3, then f = g.
171
We shall deal with Theorem A under certain value/set-sharing conditions in a sector instead of the plane C. Theorem 1. Let / € M(cr2), p{f) = oo, and <5(oo, / ) > 0. Then there exists a direction argz = a (0 < a < 2n) such that for any e (0 < e < ^ ) , if a meromorphic function g e M(oi) satisfies the condition Ec(S\,f) = Ec(S1:g) and Ex(Sj,f) = Ex(Sj,g) for j = 2,3 where n > 3 and X = X(a — e, a + e), then / = g. Theorem 2. Let / € E(<j2) and p(f) = oo. Then there exists a direction arg-z = a (0 < a < 27r) such that for any e (0 < e < ^ ) , if an entire function g G Efa) satisfies the condition Ec(Si,f) = Ec{S\,g) and Ex(S3,f) = Ex(S3,g) where n e N \ { 1 } andX = X(a-e, a + e), then f = 92.
Some Lemmas
We shall prove the Theorems by using the Nevanlinna theory in an angular domain. First of all, we recall some notations and definitions. Let f(z) be a meromorphic function on the angular domain X(a, (3) = {z\a < argz < /?}, where 0 < /3 — a < 2ir. Nevanlinna denned the following notations (see [6, 7]).
AaAr f)
>
:=
T '[ih ~ S {log+ mteia)] + log+ l^ e ^l>7
Ba,f>(r,f) ••= ^j " '
log+\f(reie)\smu;(6-a)d6
[ J ex.
(Aj-%F) s i n ^--«)>
CQ,p(r,f):=2 £ K|bm|
'
m|
where u> = ir/{(3 — a), 1 < r < oo and 6 m = |6TO|elSm are the poles of f(z) on X(a,P) appeared according to their multiplicities. Ca^{r,f) is called the angular counting function of the poles of / on X and the Nevanlinna angular characteristic is defined as follows: Sa,fi(r, / ) := Aa,0(r, / ) + Ba,fi(r, f) + C Qi/3 (r, / ) . Similarly, for o ^ oo, we can define Aatp{r, l / ( / - a)), Ba<0{r, l / ( / — a)), Ca,p{r, l / ( / — a)), Sa:p(r, l / ( / — a)) and so on. For the sake of simplicity, we omit the subscript of all the notations and use the notations A(r,a), B(r, a), C(r, a) and S(r, a) instead of Aa,p(r, l / ( / - a)), S Qi/3 (r, ! / ( / - a)),
172
Ca,0(r,l/(f - a)) and Sa,p(r,l/(f properties of S(r, f) as follows.
- a)) if a ^ 00. We shall give some
Lemma 1 [7]. Let f(z) be a meromorphic function on X(a,(3). arbitrary finite complex number a, we have
Then for
S(r,a)=S(r,f)+e(r,a), where e(r, a) = 0(1) as r —> oo. Lemma 2 [7]. Let P(z) be a polynomial of degree d > 1, and f(z) meromorphic function on X(a,(3). Then S(r,P(f))
= dS(r,f)
be a
+ 0(l).
Lemma 3 [7]. Let f{z) be a meromorphic function in C, X(a,/3) C C. Then
A(r,£)+B(r,£)=Q(r,f), where Q(r, / ) satisfies i)- Q(r, f) = 0(1) as r -> oo if p(f) < oo; ii). Q(r, f) = 0(logrT(r, / ) ) as r —> oo and r £ E if p(f) = oo, where E is the set of finite linear measure. Lemma 4. Let f(z) be a meromorphic function in C, X(a,(3) C C. Then S(r, f) < C(r, f) + C(r, j)
+ c(r, j ^ j - C0(r, f) + Q(r, f),
where Q(r,f) = 0 ( l o g r T ( r , / ) ) , r fi E and E is a set of finite linear measure. C(r, f) is the reduced counting function of poles of f, the number of each distinct pole of f(z) in X D {^||^| < r} be counted only once; Co{r, 1//') is the counting function of the zeros of f but not the zeros of f and / — 1 in X (1 {z\\z\ < r}. Next, we introduce some notations for the following main lemma. Let / be a meromorphic function in an angular domain X(a, (1). We denote by C2(r,f) the counting function of the poles of / in {z € X(a,/?)||2:| < r } , where a simple pole is counted once and a multiple pole is counted twice. In the same way, we can define Ci(r, 1//).
173
Lemma 5. Let F and G be two nonconstant meromorphic functions such that F and G share l,oo CM in X(a,0). Then one of the following three cases holds: i). S(r) < C2(r, l/F) + C2(r, 1/G) + 2C(r, F) + Q(r, F) + Q(r, G); ii). F = G; Hi). FG = 1, where S(r) = max{S(r, F), S{r, G)}, Q(r, F) and Q(r, G) as defined in Lemma 3. Let / and g be two nonconstant meromorphic functions in C, and S3 = {w\wn(w + a) — b = 0}, where n(> 2) is an integer, a and b are two nonzero constants such that the algebraic equation wn(w+a) — b = 0 has no multiple roots. We denote F=r(f
+ a)t
n
c=g
(g
+ a)
(1)
b b Obviously, if Ex(S3,f) = Ex(S3,g) then F and G share 1 CM in X. In the following, we shall give some lemmas relating to F and G. Lemma 6. Suppose that E c ( { 0 } , / ) = E~c({0},g) and 6(00, f) > 0. / / F = G, where F and G are defined as (1), then f = g. Lemma 7. Let Sj (j = 1,2,3) be defined as in Theorem 1, and let F and G be defined as (1) such that Ex(Suf) = Ex(Sug), Ex(S2,f) = Ex(S2,g) and Ex(S3,f) = Ex(S3,g). If F £ G, then C(r, -j)
= C(r, -j)
= Q(r, f) + Q(r, g).
Lemma 8. Under the condition of Lemma 7, we have C(r, f) = C(r, g)<j-
(S(r, f) + S(r, g)) + Q(r, f) + Q(r, g).
(2)
Finally, we need the following important lemma in [8, Theorem VII. 3]. We first introduce some notations. Let f(z) be a meromorphic function in an angular domain Ao : | arg z\ < ao and A : | argx| < a{< ao) be an angular domain contained in Ao- Let Ao(r), A(r) be the part of Ao, A, which is contained in \z\ < r, respectively. We put
174
T>,A)=f^U, Jo r which is called as Ahlfors-Shimizu characteristics. We write the above characteristic functions of f(z) in the whole complex plain as S*(r, / ) , T*(r, / ) . Let n(r, A 0 ,a) be the number of zeros of f(z) — a contained in A 0 , where each multiple zero is counted only once. We put N(r, Ao, a) := N(r, A 0 , / = a) = f " ( r ' A ° ' a ) r f r . Ji r Then we shall give the following analogue of the second fundamental theorem. Lemma 9. Let f(z) be a meromorphic function in the complex plane. Then for any three distinct points a\, 02,13 on C, we have 3
5*(r, A) < 3 > T n ( 2 r , Ao.oO + O(logr), and 3
T * ( r , A ) < 3 ^ i V ( 2 r , A o , a i ) + 0((logr) 2 ). i=l
Finally, from [1, Theorem 1.4], we have Lemma 10. Let f(z) Then
be a meromorphic function in the complex plane.
\T(r, f) -T*(r,f)
- log+ |/(0)|| < ±log2.
Lemma 11 [9, Lemma 1.1.1]. Let g : (0, +00) —> R, h : (0, +00) —> R be monotone increasing functions such that g(r) < h(r) outside of an exceptional set E of finite linear measure. Then, for any a > 1, there exists ro > 0 such that g(r) < h(ar) for all r > ro. 3.
Proof of Theorems
We only prove Theorem 1 since we can obtain Theorem 2 by the similar proof method. Suppose that Theorem 1 does not hold. Then for any a € [0,27r), we have a constant ea € (0, n/2) and a meromorphic function g =
175
such that Ec(Si,f) = Ec(Si,g[a]) and Ex{a)(Sj,f) = Exia)(Sj,ga) for j — 2,3, but / j£ 3[Q], where X(a) := {z\\ argz — a| < ea}. First we define F and G as (1), then F and G share 1 and co CM in X. By Lemma 6, we deduce that F ^ G. Thus, Lemma 7 implies that C(r, 1//) = C(r, l/g[Q]) = Q(r, / ) + Q(r, g[a]).
(3)
Therefore, by the definition of F and G and (3) we have C2(r,-^)+C2(r,-l)+2C(r,F) < C(r, — j - ) + c ( r , — ? — ) + 2C(r, / ) + Q(r,f) + Q(r,g[a]).
(4)
Set S'i(r) := max{S'(r, f),S(r,g[a])}have
Then, from (1) and Lemma 2, we
S(r) = ( n + l ) S i ( r ) + 0 ( l ) , where S(r) = raa,x{S(r,F),S(r,G)}. Lemma 8 and (4) we deduce that C2(r,ir)+C2(r,-l)+2C(r,F)<
(5)
By the estimate (2) obtained in
(2 + ^
Si(r) + Q ( r , / ) + Q(r,g[a]). (6)
Suppose that F G = 1. From (1) we obtain / " ( / + a)SH( 3, we have from (5) and (6), Si(r) < Q(r, f)+Q(r,g[a]). Therefore, by Lemma 3, we have S(r,f)
= 0 (logrT(r,f)T(r,g [a] )),
r ? Ea,
(7)
where Ea is a set of r of finite linear measure possibly depending on a. Therefore, by Lemma 1 and (7), for any complex number a G C, we have CX(a)(r,
a) := Ca-£a,
a+ea(r,
a) = O(logrT{r, f)T(r,g[a]))
, r # Ea. (8) On the other hand, we define X(a)\ = {z\ \ argz — a\ < ea/2}. Noting that if an a-point bm = |&m|el6>m of / is in X ( a ) i , sinw(# m — (a — ea)) > sin(a;-^) = \/2/2
176
since w = wa := n/ (2eQ) (> 1), we have
Cx{a)(2r,a)
=2
£
{ - ± - - M l } s i n J ^ - (a - ea))
K|bml<2r \0m-Ot\<Ea
1<,tr<.
W w»}
\9m-a\<ea/2
_
P2r
^fn(2r,X(a)i,g)
\
(2r)"
n(t,X{a)ua)
^A
^+!
"(27F n ( 2 r ' x ( a ) l ' a : ) + (27p/ > ^{
n ( r
^:'a^2r-r)+n(r,X(a)
l l
n
M( Q )^) r l l l t }
a)r-1(2r-r)|
> V^ 2n n( r( ,r J, X r ((aa))l1j ,f al )) {<^! l^ T r r - + r It follows from (8) that n ( r , X ( a ) i , a ) = 0 ( r w logrT(2r,/)T(2r, f f w )), r £ £ Q .
(9)
If we identify the interval [0,27r) with the unit circle and (a — e Q /4, a + e Q /4) with the corresponding open arc on the unit circle. Then since the unit circle is compact and
[0, 2n)C (J ( a - ^ ,
a+'-f),
a€[0,27r)
we can choose finite many coverings (ai — £ Q l /4, a.\ 4- £ a i / 4 ) , (a2 — £a 2 /4, a 2 + £a 2 /4), • • •, ("fc - £a fc /4, afc + £Qfe/4) of the interval [0,27r). Therefore, using Lemma 9 and (9), for any three distinct complex numbers Oj, j = 1,2,3, we have
Sr{r,f)<^S*(r,Ai,f) i=l k
3
< ^ { 3 ^ n ( 2 r , A t , f = a , ) } + O(logr) t=i
= 0(r
j=i fi
logrr(4r,/)T(4r)S[Q])), r £ £ 0 ,
177
where At = A ( a . _ £ Q . / 4 > and So :=UJ=.E Qi . Thus,
Qj+£ai/4)
(i = l,---,k),
fi
:= m a x { w ( a i ) , . . . ,w(a f c )}
T*(r,/) = 0(rn+1logrT(4r,/)T(4r,5[a])),
r £ £0.
(10)
By Lemma 10, it follows from (10) t h a t T(r,f)
= 0{rn+1
l o g r r ( 4 r , / ) T ( 4 r , < 7 [ a ] ) ) , r $ E0.
It follows from Lemma 11 t h a t there exists ro > 0 such t h a t for all r > TQ, we have T(r,f)
= 0 ( r n + 1 logrT(5r-,/)T(5r,3[Q])),
which contradicts p ( / ) = co since ,. log log T ( r , / ) ^ hmsup : < oo and r—oo
logr
hmsup r^oo
loglogr(r,ffM) : — < oo. log r
This completes the proof of Theorem 1. References 1. W. K. Hayman, Meromorphic Functions, Clarendon, Oxford, 1964. 2. H. X. Yi and C. C. Yang, Uniqueness Theory of Meromorphic Functions, Pure and Applied Math. Monographs No.32, Science Press, Beijing, 1995. 3. F. Gross, On the distribution of values of meromorphic functions, Trans. Amer. Math. Soc, 131 (1968):199-214. 4. F. Gross, Factorization of meromorphic functions and some problems in Complex Analysis (Proc. Conf. Univ. Kentucky, Lexington, KY, 1976), Lecture Notes in Math., 599 (Springer-Verlag, Berlin, Heidelberg, New York, 1977), pp. 51-69. 5. H. X. Yi, On a question of Gross concerning uniqueness of entire functions, Bull. Austral. Math. Soc, 57 (1998):343-349. 6. R. Nevalinna, Uber die Eigenschaften meromorpher Funktionen in einem Winkelraum, Acta Soc. Sci. Fenn., 50 (12)(1925):l-45. 7. A. A. Gol'dberg and I. V. Ostrovskii, The Distribution of Values of Meromorphic Functions (in Russian), Izdat. Nauk. Moscow 1970. 8. M. Tsuji, Potential theory in mordern function theory, Maruzen Co. Ltd, Tokyo, 1959. 9. I. Laine, Nevanlinna Theory and Complex Differential Equations, De Gruyter, New York, 1993.
T H E C O N V E R G E N C E OF LAPLACE-STIELTJES TRANSFORMS*
L U O XI a n d SUN D A O C H U N Department of Mathematics South China Normal University Guangzhou, 510631 P. R. China E-mail: [email protected]
In this paper.we study the convergent properties of Laplace-Stieltjes transforms and obtain some completely different results.
Keywords: Laplace-Stieltjes transform; abscissa of convergence; abscissa of uniform convergence; abscissa of absolute convergence. 2000 Mathematics
1.
Subject Classification:
30D35
Introduction
Yu Jiarong [1] combined Valion [3]and Knopp [4]'s ideas,and extended the Cauchy-Hadamard formulas of powder series to the Laplace-Stieltjes transforms. Yu Jiarong [5] had the corespondent results about the double Dirichlet series and double Laplace-Stieltjes transforms. The results about Laplace-Stieltjes transforms were relied on a sequence of {A„}, which was useful to apply some results about Dirichlet series. In the paper,we manage to obtain some new results which are not relied on {A„}, but just relied on a constant.
"The Project supported by the National Natural Science Foundation of China. 178
179
2. Main Results We define Laplace-Stieltjes transform as F(s) = / e~syda{y), Jo Where s = a + it is a complex variable,and a(y) is a real or complex function and of bounded variation on any interval [0, X], X £ (0, +oo). We define the abscicca of convergence and abscicca of uniform convergence and abscicca of absolute convergence respectively as a^ = inf{oo;F(s) is convergent for
e-s°yda(y)\
< |
Ju
for u> M. Then, for all u\, U2 > M, e-aoyda(y)\<\
\
Soy
e-
da(y)\
+\ Ju2
da{y)\
Ju\ Ju\ Denote Iu(x; so) = f* e~$aVda{y). For all r £ [0, §), taking s £ Er, since rx
/
<2
£ +
2
= £
-
px
e-syda(y)
= /
J U
= fX e-l-'°»dlu(y;
£
SoV
e-
e-Soye-^s-So">vda{y)
JU
s0) = Iu(x; s 0 ) e - ( s - s ° ) x - f Iu(y; a 0 )de- ( -" , ) w ,
Ju
Ju
taking u > M,x > u, then we have | f e-syda(y)\ Ju
< \Iu(x; s0)\e-^-^x
+ f' \Iu{y;s0)\e-^-^y\s Ju
- s0\dy
180
<
ee
-(a-a0)x
+ e |i_!°.|[ c -(a-<*)* + e - ( < r - a 0 ) „ ,
(7 — (To
Therefore, F(s) is uniformly convergent on £?r. According to 1), we can obtain 2) easily. Theorem 1 < where p(x) =
F
- p lnp(a;) = lim ^ - , x—>+oo
a;
sup |/ x da(y)|, K is a positive constant. x
Proof
We write A=
lim x^+oo
l
a;
-^^-,qx(y)=
fZda(y).
yx
At first, we consider that A is finite. On one hand, we take cr0 > A,e=
ao - A — - — >0,
then there exists^ > 0, such that ^ £ i < A + ^f^ = ^ ^ f o r x > X. That is p(x) < e x£a 5 — for x < X. Taking x < X, x < z < x + K, we have
| [Zda(y)\
Since rx+K
/
e-y°°da(y)\
rx+K
= | /
JX
e-v°dyqx(y)\
JX
rx+K —
\e~(x+K)
x(x
qx{y)e-yaody\
+ K)+a0 JX
< e-(x+K)aoex^Y^-
-f exZs^\e~xa°
— e~(x+K'*'To\
an — A
when a0 < 0; K(a0) = 1, when a0 > 0.
/•+oo
+°°
rX+nK
e~y°ada{y)\ < V | /
\ JX
n = 1
JX+(n-l)if
e~^da{y)\
181
=
^(
_ ^ = tf ( a 0 ) e ^x ^£ a
1 — 3 x < +00. 1 — e~ A 2 Now, we see F(s) is convergent at s = aoBy Lemma 1, F(s) is convergent for a > A. So af
°Z>A. Suppose F(s) is convergent at s = ao, then there exists a positive real number M , such that for all £1,2:2 € (0,+co), e-
I / Jx\
Denoting e-°°yda(y),
P*(*;*o) = f Jx
we have \Px{y\
I f da(y)\ = I f JX
= If
JX
= \px(z;a0)ea°z
e°°vdpx(y;ao)\
- a0 f
Jx
Jx 02
< Me* K(a0) = 2MeKa°, So
eaaVpx{y;<70 )dy\
z
+ M\e°°
x
- e"° \ <
K(a0)ea°x.
when a0 > O.K(a0) = M, when
If Jx
da(y)\ < K{aQ)e°0X.
182
That is p(x) <
K(a0)e°°x.
Now we obtain —: lnp(:r) FV ; lim < a0. x-»+oo
x
Since
o-l < A. When A = +00, by the above proof, we know a£ > +00. That is,crf = +00. When A = — 00, for all a < 0, there existsX > 0, such that <e2*x
f da{y)\ Jx
for x > X, z > x. For all x > X, we have x+K rrX-f/\
e-y°da(y)\
/ Jx
< e2axe-cr(-x+K)
+ e*™[e-°{*+K) _ e -»*]
< 2 e 2cr:c e - CT ( x +^)
=
2e2ax~Kcr
What is more, /
e-" ff da(y)|
e-«"da(y)|< T \ /
•/AT
JX+(n-l)K
n = 1
2e- KCT T e-^+t"- 1 )*] = 2 e - K f f + ^
^ - < +00.
n=l
It shows that <7,f < <J. SinceCTis arbitrary, a^ = —00. Similarly, we obtain L e m m a 2 If F{s) is uniformly convergent on the line a = ao, then F(s) is uniformly convergent on the right half plane a > ao. Proof From the condition, we see, for all e > 0, there exists M > 0, such that (•+00 r+00
I/ Ju
0+it)y
e-(°
da{y)\
<
183
for u > M and t € R Then, for all u\, ui > M, IC
e-^+«)»da(!/)| < | J*00
e-^+^yda(y)\
+\C2°°e-{!T0+it)vda(y)\
+ 62=£-
Denote Iu{x; a0 + it) = / J e - ^ + ^ o f c ^ y ) . Take s = a + it, since / J e- s ^da(j/) = / * e-(' ro+it ^e- (
+ it) - / * J u fo; a 0 +
it)de~^-"°>»,
taking u > M, a; > u,
it)\e^"-^x
+ Iu \iu(y;o-0 + it)\e-^-^y\a -
< e + 2e = 3e. Therefore, F(s) is uniformly convergent for cr > CoTheorem 2 F
-^x—»+oo
where n{x) =
lnn(x) a;
| / e~ltyda(y)\,K
sup
is a positive constant.
t£( — oo,+oo),x
Lemma 3 If F(s) is absolutely convergent at the point s = ao, then F(s) is absolutely convergent for a > aoTheorem 3 P a„ =
where m(x) = J
—rhm x—>+oo
hirn(x) a;
\da(y)\, K is a positive constant.
184 References 1. J. R. Yu, The Borel Line Of Entire Function Defined By The Laplace-Stieltjes Transform, Maths Journal, 1963:471-484 2. J. R. Yu, Dirichlet Series And Random Dirichlet Series, Science Press,1997. 3. G. Valiron, Entire functions and Borel's directions, Proc. Nat. Acad. Sc, U.S.A., 20(1934):211-215. 4. K. Knopp, Uber die Knovergenzabscisse des Laplace-Integrals, Math Zetts, 54(1951):291-296. 5. J. R. Yu, The convergent property of double Dirichlet series and double Laplace-Stieltjes transforms, Jounal of Wuhan university (science), 1962:1-17.
A N E W D E C O M P O S I T I O N FOR T H E H A R D Y SPACE ON DOMAINS*
ZENGJIAN LOU Department of Mathematics Shantou University Shantou, Guangdong, 515063, P. R. Email: [email protected]
China
We give a new atomic decomposition for the Hardy space on domains.
Keywords:
Hardy space; atomic decomposition; domains.
2000 Mathematics
Subject Classification:
42B30
1. Introduction Let
= sup \
belongs to L ^ R ^ ) ([9]). If this is the case, define ll/llwi(RN)
=
\\M(f)\\LHM.N)-
The theory of Hardy spaces on domains 0 C R w was developed, among other places, in [7] (arbitrary domains) and in [3] (bounded Lipschitz domains). Let fi be an open domain in R™. A function / on fi belongs to "This work was supported by Natural Science Foundation of Guangdong Province (Grant No. 032038), National Natural Science Foundation of China (Grant No. 10371069) and SRF for ROCS, State Education Ministry. 185
186
H\ (Cl) if the function F denned by
f/W, if* en; iixenc.
[0,
belongs to W^R") with | | / | | w i ( n ) = | | - F | | « I ( R « ) . A Lebesgue measurable function a is said to be an Hl(Q)-atom if there exists a cube Q with 8Q C fi and supp a C Q such that a satisfies the moment condition / a(x) dx = 0 JQ
and the size condition
MU'(R") < \Q\~l/2Those atoms we denned are different from the ordinary ones, since supports of them are away from the boundary of the domain J7. This is convenient to use when we deal with problems related to the boundary dCt. In this note using an idea in [6], we prove that any function in 7i\(Q) can be decomposed into ?^(fi)-atoms. 2. Theorem and its Proof Theorem 1. Suppose Q. is a Lipschitz domain in M.n. A function f on Q, belongs to Hi (Q) if and only if it has an atomic decomposition oo
/ = / yAfcafc, k=o where the a^ 's are Hi (ft)-atoms and SfcLo l^fcl
<
°° -
For the proof of Theorem 1, we need the following lemma of Necas in [8, Lemma 7.1 in Chapter 3]. Lemma 1. Let ft be a bounded Lipschitz domain in R™. Then the divergence operator is a continuous map from HQ(Q,M.N) onto Ll(Q) = {/ G L 2 (ft) : JQ f dx = 0}, that is, there exists a constant C depending only on the domain fi and the dimension N such that for any f e LQ(D.), there exists ip in the Sobolev space WQ' (£2,1^) such that f = div
187
Proof of Theorem 1. We only prove the necessity. From Theorem 3.2 in [3] (see also, [2], [5], and [1]), any / G H\{0) can be written as oo
fe=o where supp otk C Qk, cubes in fi such that / Q au dx = 0, Ha/cH^Q^ < \Qk\-1/2,
and OO
EN
For convenience we drop the subscript k temporarily. Applying Necas' lemma to the function a — au, one finds that there exists (p € W0' (Q, RN) and a constant C independent of cp and Q such that a = div ip and II V P\\LHQ,UL") < C||"I|L 2 ((3)-
Applying a Whitney decomposition with respect to the boundary dQ, Q can be decomposed into a family of subcubes: Q = U £ o Qi s u c n * n a t 8Qi C Q and |Q| = E S o \Qi\- Using this decomposition we construct a smooth partition of unity YliLo Vi(x) = 1 in Q, where r]i(x) = 1 in Qf, r]i(x) = 0 outside 2Qi and | y 77* | < C / ( Q i ) - 1 . We have
= Et~0
div
( w ) =• ESo 7 "^'
a n d Ti = where a* = [ z Q d 1 ^ fa^lL^) l 2 ^ l 1 / 2 H d i v (W)IIL'(2Q«)It is obvious that cc, is an Til(Q)-a,tom supported in 2Q, with 8Qi C fi. So Theorem 1 is proved if we can show that there exists a constant C independent of a and Q such that E ^ o Ti — ^ < °°- Write
E , ~ o ^ = E £ o I2^| 1 / 2 ||div ( w l I U ^ ) < 2 1 / 2 E,~o |2Qi| 1 / 2 (||^div ^|| L 2 ( 2 Q i ) + || V Vi •
=:/ + //. Note that 0 < rji < 1. By the Cauchy-Schwartz inequality, we have ^<21/2ESol2Qi|1/2||div^||L2(2Qi) < 2 1 / 2 (E,=o I2ftl) 1 / 2 ( E S o f2Qi l«l2 ^ )
V 2
188
We now deal with the second p a r t II. Note t h a t | y rji\ < C / ( Q i ) _ 1 < Cd(x,dQ)~1(x £ 2Qi) and Hardy's inequality (see, for example, [4, Chapter 1, Section 5]) give 1/2
n
dx
yr/i-V 2
V>(x)
d(x,8Q) 2
d(x,dQ)
>V2 dx
\!/2
dx
where C does not depend on Q and a, T h u s each a t o m a*; can be decomposed into ? ^ ( f i ) - a t o m s a\ and the supports of these atoms can be away from the boundary dO,.
References 1. P. Auscher, E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of RN, J. Fund. Anal. 201(2003):148-184. 2. D. C. Chang, The dual of Hardy spaces on a domain in R™, Forum Math. 6(1994):65-81. 3. D. C. Chang, S. G. Krantz, E. M. Stein, Ttp theory on a smooth domain in RN and elliptic boundary value problems, J. Fund. Anal. 114(1993):286-347. 4. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989. 5. A. Jonsson, P. Sjogren, H. Wallin, Hardy and Lipschitz spaces on subsets of R n , Studio Math. 80(1984):141-166. 6. Z. J. Lou, A. M c Intosh, Hardy spaces of exact forms on Lipschitz domains in R", Indiana Univ. Math. J. 53(2004):583-611. 7. A. Miyachi, Hp spaces over open subsets of R n , Studia Math. 95(1990):205228. 8. J. Necas, Les methodes directes en theorie des equations elliptiques, Masson et Cie, Eds., Paries; Academia, Editeurs, Prague 1967. 9. E. M. Stein, Harmonic Analysis, real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, New Jersey, 1993.
T R A N S V E R S A L I T Y O N C O O R D I N A T E D MANIFOLDS
TSOY-WO MA School of Mathematics and Statistics University of Western Australia Nedlands, W.A., 6907, Australia E-mail: [email protected] We develop elementary properties of locally compact morphisms from coordinated manifolds transversal to submanifolds of coordinated manifolds as extension of regular values and submersions. Transversal submanifolds are treated as a special case. Examples are given at the end.
2000 Mathematics
Subject Classification:
57N75, 58B15
1. Introduction The difficulty of many definitions of infinite dimensional differential calculus beyond Banach spaces is summarized in 8 . Global analysis 5 based on convenient spaces and also products of generalized functions 2 in terms of bornological vector spaces make use of differentiable maps which need not be continuous. Quantum probability is used to study infinite dimensional analysis u . However, we prefer to work in infinite dimensional complex analysis on traditional locally convex spaces. In contrast to various definitions of holomorphic maps given in 3 , all our holomorphic maps must be locally bounded. Our first application is to provide a simple framework for products of distributions 10 . With holomorphic locally compact perturbations of continuous linear maps, we initiate a primitive theory of infinite dimensional holomorphic manifolds 8 . A complex quasi-complete separated locally convex space E is called a coordinate space if every compact set is contained in a closed vector subspace on which there is a continuous norm. This better definition is equivalent to 7 in terms of a coordination {Ej : j £ A} if we let Ej be the smallest closed vector subspace containing j for each j in the family A of all compact subsets of E. Coordinate spaces are important because the inverse 189
190
mapping theorem is available to enrich the primitive theory on coordinated manifolds which are holomorphic manifolds modelled on coordinate spaces. Immersions, embeddings, submersions, regular values, regular points, level sets, locally linear maps and subimmersions have been established in 9 . This paper extends these results to the context of transversality similar to the treatment in 4 for finite dimensional manifolds and in 1,e for Banach manifolds but the generalization of Whitney topology 4 will be considered later. We shall use the notation of 8 and 9 without any further explanation. 2. Transversal Maps Let M, TV be holomorphic manifolds modelled on coordinate spaces E,F respectively, / : M —> TV a locally compact morphism and Y a submanifold of TV. Then / is said to be transversal to Y at m € f~1(Y) if the following conditions hold. (a) df(m)(TmM) + TnY = TnN where n = f(m), (b) df{m)~l{TnY) splits in TmM. The first condition is also called the transversal equation. The map / is transversal to Y if it is transversal to Y at every point in f~1(Y). Clearly if / ( M ) is disjoint from Y, then / is transversal to Y. We shall assume that Y is modelled on a split subspace G of F and H = FQG is any topological complement. Lemma 1 / / every point of Y is regular value of f, then f is transversal toY. Proof Take any m £ f~1(Y). Since df(m)(TmM) = TnN, the transversal equation holds. Let (W,ip) be an adapted chart of TV for Y at n = f(m), that is ip(W n V ) = ip(W) D G. By Local Submersion Theorem 9, choose a chart (V,tp) of M at m such that tp(V) C W and ipf(p~1(x) = J(x) for all x G
E1®E2 1
~[Jr (G)®Jr1(H)]®E2 -Jr1{H)®[J^l(G)®E2] =
Ji1(H)®J~1(G)
191
~ [dipim)}-1 J^{H) ~
0 [Mm)]'1
J_1#(n)(T„r)
[dp(m)]-1Ji1{H)®d(il>-1J
~ [Mm)]-1 J^\H) e 4f(m)-1(r„y), d / ( m ) - 1 ( T „ y ) sp/ite in TmM.
Consequently, f is transversal to Y.
Submanifold Lemma Every submanifold is locally the inverse of a regular value. More precisely, for every adapted chart (W,tp) at any point n G Y, there is a submersion g on W into H such that g(n) = 0, WDY = g-^O) anddg(n)(TnN) = H. Proof Let IT : F = G ® H -> H be the projection. Then g : W -> H given by g(w) = TT[IP(W) — ip(n)] for every w G W is a locally compact morphism with g(n) = 0 and dg(n)(TnN) = dir(0)dip(n)(TnN) = TTF = H. Since ker(n) = G splits, g is a submersion on W. In particular, 0 is a regular value of g. Finally, take any y G W. Then y G W D Y" iff ^(y) G ^ ( W n V") = ^>(W) n G iff 4>(y) G G iff V(y) - V>(») e G iff $(y) = 0 iff y G <j_1(0). This completes the proof. The following elementary observation is useful. Let A : E —> F be a continuous linear map and B : F —> if be a continuous linear surjection. Then BA is surjective iff AE + ker(B) = F. We also have ker(BA) = A " 1 [ker(B)] and E/ker(BA) ~ £ A ( £ ) . Lemma 2 Lei W be an open compact morphism on W into Wf\Y = g~l{Qi). Suppose that transversal to Y at m G / _ 1 ( n )
neighborhood of n G Y and g a locally a coordinate space with g(n) = 0 and g is submersive on W(lY. Then f is iff the composite gf is submersive at m.
Proof Since g is submersive on g _ 1 (0), n is a regular point of g. Hence dg(n) : TnN -> H is surjective and we have ker[dg(n)] = Tn(WC\Y) = TnY. Clearly, d(gf)(m) = dg(n)df{m) is surjective iff df{m){TmM)
+ ker[dg(n)} = TnN
iff the transversal equation holds. We also have
ker{d(gf)(m)\ = df {m)'1 {ker[dg{n)}} = Therefore ker[d(gf)(m)} the proof.
splits iff df(m)~1(TnY)
d f ^
1
^ ) .
splits. This completes
192
Transversal Mapping Theorem Let f : M —> N be a locally compact morphsim transversal to a submanifold Y of N. Then (a) X = f~1(Y) is a submanifold of M. (b) TmX = dfimy^TnY) for allm&X. (c) TmM 0 TmX and TnN Q TnY are topologically isomorphic. (d) IfY has finite codimension in N, then codim X = codim Y. Proof Take any m £ X. Then n = f{m) G Y. Choose W and g by Submanifold Lemma 2. Let V be an open neighborhood of rri with /(V) C W and let h = f\V. Since / is transversal to Y, gh is submersive at every point of V D / _ 1 ( n ) , or m is a regular point of gh. By Level Set Theorem 9 , X DV = (<7ft)-1(0) is a submanifold of V and we also have Tm(X n V) = ker[d(gh)(m)]. Hence X is a submanifold of M. Clearly, we have TmX = Tm(X n V) = ker[d(gh)(m)} =
ker[d(gf)(m)]
=
df{m)-\TnY).
Part (c) follows from the calculation: TmM 9 TmX ~ ~
TmM/ker[d(gf)(m)] dg(n)df(m){TmM)
= dg(n)[df(m)(TmM) = dg(n)TnN
= H,
+ kerdg{n)\ by Submanifold Lemma.
Part (d) follows by taking dimensions of (c). This completes the proof. Theorem 1 Let P, M, N be holomorphic manifolds and h : P —> M, f : M —> N locally compact morphisms. Suppose that f is transversal to a submanifold Y of N. Then h is transversal to the submanifold X = f~1(Y) of M iff fh is transversal to Y. Proof Take any z G (fh)~1(Y). Let m = h(z) and n = f(m). Construct (W, tjj) and g by Submanifold Lemma Choose an open neighborhood V of m with f(V) C W. Then (gf)(m) = g(n) = 0. Pick any v £ V. Then v G (9/) _ 1 (0) iff (gf)(v) = 0 iff f(v) G g-\0) = WHY i$v G f-^Y) = X iff v G V n X. Thus V n X = (ff/) _1 (0). Since / is transversal to Y, gf is submersive at every point of V n X. Therefore h is transversal to X iff
193
(gf)h = g(fh) is submersive at every point z S X iff fh is transversal to Y. 3. Transversal Submanifolds Let X, Y be submanifolds of a holomorphic manifold M modelled on a coordinate space E. Clearly the inclusion map / : X —> M is a locally compact morphism and its derivative df(m) : TmX —» TmM is an inclusion map for every m £ X. Now X is said to be transversal to Y if / is transversal to Y, that is for all m G X D y , (a) TmX + TmY = TmM, (b) TmX n T m y splits in TmX, or equivalently in TmM. Theorem 2 Lei X, Y be transversal submanifolds of M. Then (a) X n y is afco a submanifold of M. (b) For every meXnY, we have Tm(X nY) = TmX n T m y . (c) codim[Tm(X n y)] = codim(TmX) + codim(TmY) if the right hand side is finite. Proof Only part (c) requires a proof. For convenience, write F = TmM, A = TmX and B — TmY. Since the restriction a to B of the quotient map F —> F/A is continuous linear surjective, we have F/A ~ B/ker(a)
= B/(A n B).
By F = A + B, we obtain F / A n B = A/A(1J3 + -B/ADB. Clearly this is an algebraic direct sum. Therefore we get codim[Tm(X
n y)] = dim F/A n S = dim A/A HB + dim B/A n S = dim F/B + dim
F/A
= codim(TmY)
codim(TmX).
+
4. Examples Let En be the set of sequences x = {x\, x%, • • • ) with Xj = 0 for all j > n. As a finite dimensional vector space, each En is equipped with a unique locally convex topology. The strict inductive limit E = U^Li En consisting of all finite sequences is a coordinate space. It is well known that E is not metrizable and hence not a Banach space.
194
If Aj is an open subset of then the set
n
oo
j=1
Aj = {x£E:
Xj
£ Aj , Vj}
is open in E. In fact, take any a £ A = YiTLi A r Choose Sj > 0 such that aj + M(6j) C Aj where M(5j) = {t £ C : \t\ < Sj}. Then the convex balanced absorbing set V = YljLi ®(<^') is a O-neighborhood of E such that a + V C A. Therefore A is open in E. In particular, the set M = {x £ E : \XJ\ < l/2 :? } is open in E. Therefore it is a coordinated manifold. For morphisms / between open sets of coordinate spaces, we identify the differential df(m) and the total derivative Df(m). The function f on M given by J v
;
^ j = i 2 - Xj
is holomorphic. In fact, each composite function x —> Xj —> Xj/(2 — Xj) is holomorphic on M. Since \XJ/(2 — Xj)\ < \XJ\ < 1/2J, the uniform limit / is also holomorphic. For each h = (hi, /12, • • • ) £ E, we have
Df(x)h= jtf(x
T-^uu
+ th) t=o
^
= 1
hj
( " ^)2 2
Since Df(x) ^ 0, it is surjective onto C Because Df(x) is a continuous linear form on E, its kernel splits. Therefore every point in M is regular. The level set X = / _ 1 ( ° ) i s a submanifold of M. Consider the locally compact map K on M into £2 given by K(x) = (zie X3 -x2,xieX3
+ z 2 ,0,0, • • •)
Then we have DK(x)h=
( ^ e 1 3 -h2
+ h3xieX3,hleX3
+h2 +
h3xleX3,Q,Q,---).
Because DK(x) carries E2 onto E2, the linear map DK(x) : -E —> E2 is surjective. Since the kernel of DK(x) is a closed vector subspace with finite codimension, it splits. Therefore level set Y = if _ 1 (0) is also submanifold of M where 0 is the zero vector of E. Take any m = (mi,rri2, m^, • • •) & X n Y. From K(m) = 0, we have mi = m2 = 0. For each h £ E, we need to find a £ TmX = ker[Df(m)\ and b £ TmY = ker[DK(m)} such that h = a + b. To satisfy DK(m)b = 0, let 61 = b2 = 0 and bj = hj for all j > 3. Set ai — hi, a2 = h2, aj = 0 for all j > 3. Solve for 03 from njv
\
ai
.
a2
03
n
195 Define b3 = h2 - a3. Hence TmX + TmY = TmM. Next, TmX n TmY is a closed vector subspace with finite codimension, it splits in TmM. Consequently X, Y are transversal submanifolds of M. T h e left shift L defined by L(x) = (x2,x3,---) is a continuous linear m a p on E. Hence g = L + K given by g(x) = (xieX3,xieX3
+ x2 +
x3,X4,x5,-••)
is a locally compact morphism from M into E. Furthermore, we have Dg(x)h
= L(h) + 3
= (fixe*
DK{x)h + h3xieX3,hieX3
+ h2 + h3xieX3
+ h3, /i 4 ,
h5,---).
For each k G E, define h G E by hi — kie~X3, h2 = k2 - k\, h3 — 0 and hj = /Cj_i for all j > 3. T h e n Df(x)h — k. Hence I?5(a;) : E —> £ is surjective. Since fcer[£)g(a;)] is finite dimensional, it splits. Therefore every point in M is regular. T h e level set Z = g r _ 1 (0) is a submanifold of M . Take any m € X D Z. From ff(m) = 0, we have m2 + m3 = 0 and m i = rrij• = 0 for all j > 3. Coupling with / ( m ) = 0 gives m2 = m3 = 0. Hence X n Z = {0}. Let m = 0. For each h £ E, we have a G T m X , b G T m Z and ft. = a + b where a2 = h2, b2 = 0, a,j = 0 and bj = hj for all j ^ 2. T h u s TmX + TmZ = TmM. Because TmX D TmZ is finite dimensional, it splits. Therefore X, Z are also transversal submanifolds.
References 1. A. Abraham, J.E. Marsden and T. Ratiu, Manifolds, tensor analysis, and applications. Second edition, Springer-Verlag, 1988. 2. J. F. Columbeau, Differential calculus and holomorphy, North Holland Math. Studies 64(1982). 3. S. Dineen, Complex analysis in locally convex spaces, North Holland Math. Studies 57(1981). 4. M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, 1973. 5. A. Kriegl and P.W. Michor, The convenient setting of global analysis, Math. Surveys and Monographs, Amer. Math. Soc, 53(1997). 6. S. Lang, Fundamentals of differential geometry, Springer-Verlag, 1999. 7. T. W. Ma, Inverse mapping theorem, Bull. London Math. Soc. 33 (2001): 473-482. 8. T. W. Ma, Holomorphic manifolds on locally convex spaces, accepted by Analysis in Theory and Application, 2004. 9. T. W. Ma, Locally linear maps on coordinated manifolds, accepted by Proceedings, 12-th Finite or infinite dimensional complex analysis, Tokyo, 2004.
196
10. T. W. Ma, Infinite dimensional complex analysis as a framework for products of distributions, accepted by Bull. Institute of Math., Academia Sinica, 2005. 11. M. Schurmann and U. Pranz, Quantum probability and infinite dimensional analysis from foundations to applications, World Scientific, 2005.
O N LOGARITHMIC (a,/3)-BLOCH SPACES*
XIAOGE MENG Department of Mathematics Shantou University Shantou 515063, Guangdong, P. R. China E-mail address: [email protected]
Define the logarithmic (a, /3)-Bloch space and characterize the logarithmic (a, 0)Bloch functions by their Taytor coefficients. The relationship between logarithmic (a, /3)-Bloch space and Cesaro means is given.
1. Introduction Let D = {z € C : \z\ < 1} be the open unit disk in the complex plane C, T = {z € C : \z\ = 1} its boundary, and denote H{D) the class of analytic functions on D. Let L1 be the space of Lebesgue integrable functions on D and L\ be the closed subspace of L1 consisting of analytic functions. For / S L1, the Toeplitz operator Tf : L\ —* L\ and Hankel operator Hf : L\ —> L1 are denned as follows:
Tf(9) = P(fg),
Hf(g) =
(I-P)(fg),
where P is the usual Bergman projection and / is the identity operator, respectively. Attele [1] proved the following theorem. Theorem A (i) If / is in L\, then Hj is bounded on L\ if and only if sup(l-|z|2)|/»|log-^
1 — |Z|
1
(ii) If / in L is real-valued and harmornic, then Tf is bounded on L\ if "This research is supported in part by the National Natural Science Foundation of China (No. 10371051, No. 10371069) and the NSF of Guangdong Province of China (No. 04011000). 197
198
and only if / is bounded and l
sup(l - M,22) v| V- / ( z ) | log — ^
< oo.
For 0 < a < oo, —oo < j3 < oo, we define the logarithmic (a, /?)-Bloch space, consisting of all functions / £ H(D), for which l l / l l i o g B ( a , « s u p ( l - ^ | 2 n / ' ( 2 ) | Lg-^—) l
\
z£D
- \z\
/
denoted by logB^^y By choosing different a and j3, it will be B, Ba, log-B, l o g B a respectively. A function / £ logB^^ is said to belong to log£0(a,/3) if
i jm i (iH^n/'wi(i«g r ^j 5 ) Note that || - liiog s ( a
0)
/,
= o.
defines a seminorm while the natural norm is given
by H/Hiog = 1/(0)1 + ll/H iogB(Qi<3), which makes log -B(Q,/3) into a Banach space. 2. The coefficients of logB( Qi/ g) functions Theorem 2.1. in H{D).
Let 0 < a < oo, —oo < /? < oo and / ( z ) = X^°=i an^™
logB (Qij8)> i/ien n 1 - Q |a„|(logn)' 3 = O(l)
i)Iff£
ii) If f € logBO(a,0),
1 a
0
then n - \an\(logn)
= o(l)
(n -+ oo); (n-> oo).
Proof, i) Assume / £ log£?( ai/3) . Let z = r£, 0 < r < 1, £ £ T. Using Cauchy's integral formula, we obtain \an\ = 7, 2n-K
i/l/M)|r'-»l*|
< ^
IT
0
,.1-n
^ / T d - 2 r i / ' K ) i ( i o g r ^ ) ^ ^ ( l o g ^ ) ' I«I "
n(l-r 2 )«
V
!-r:
199
Choosing r = (1 - ^) 2, n > 2, we have a 1
K l < ||/||logB(Q, 0) n -
I1 -
(logn)"".
Since lim
1
n->oo \
)
=e",
n ,
we get n1-a\an\Qogn)ft=0(l),
n -» 00.
ii) Assume / € logB 0 (a,^)- Then for any e > 0, there exists / € (0,1), and for any r € (J, 1), we have
(l-r^l/CrOl^og^)^^!For all n s M, we have ,
|a
. 1 "' = 2 ^
JVKXO1-"!^! 0
„l-n
log-
Kl.
Consider r = (1 — ^ ) s , n > 2. Since lim
e2,
1- -
71—»oo \
n
for the above e, there exists J V e N , with N > ^ n >N 1,l-n
(1
n
> 0, such that for all
1
) ~ < e5 + 1 < e + 1 .
Notice that r = ( l - i ) 3 £ (Z, 1). For all n > N a l \an\ < ——n - (\--\ e+1 V
".
'
(logn)-/3
This means that n
a
\an\(\ogn)P
= o(l), n —» 00
200
3. logB( Qi/ 3) space and Cesaro means For given a function f{z) = X^Lo 0 " 0 ™
e
^{D),
we write
n
= ^2akzk,
sn(f)(z)
n =
0,1,2,...,
k=0
and
*«(/)(*) = t
s
-^r = £ ( 1 - ^ 1 ) «** »=0. L 2> ••••
for the partial sums and the Cesaro means of the power series for / . n > 0 and z £ D, then (see [2]) 2n JT where
fc=-n
V
y
is Fejer kernel with the properties as follows (see [3]): 1)
Kn(0>0;
2) &/ r tf n (0|de| = l. Theorem 3.1. Let 0 < a < 00, —00 < /? < 00, and / ( z ) = Y^Lo H(D). Then f G logB(Q)/a) (f and on/?/ if sup{|k„(/)||iog B(Q , 0) : n e NU {0}} < 00. Moreover, sup{|M/)||iogfl ( o .„
:
» G NU {0}} = 11/11
Proof. Since
= l. J^Kn(Of(zom, we get
M/))'(*) = ^j
Kn{e)f\zom\.
a
nZ
201
Assume that / £ logB(a
\
1 J- -
^
\z\
< J_ /" su P (i-|z| 2 r|/'(^)l ( i ° g r n ^ ) * » < « ! ^7T JTz€D
\
>--\Z\
)
= \\fhoSB(a,m^lTKn(om = ll/l|logB(a,0)This implies that
fr(z) = f(rz) / oo
= z-\l
- rf
\
/ oo
£ ) ( n + l)r n \n=0 oo
£) /
> na r
" "- 2 "
\n=0
n + 1- k \
n=0
\fe=0
z-\l-rfY^{n+l)za'n{f)^>n oo
( l - r ) 2 ^ ( n + l)a;(/)(z)r". n=0
Hence,
|/;(*)| <(l-r) 2 5>+l)K(/)(z)|r". n=0
„
202
Consequently,
(i-i,i 2 )«i/;(.)i(io g I -^) < (1 - |z| 2 )«(l - r) 2 ( f > = (1 - r ) 2 ^(n n =o
+ l)\a'n(f)(z)\rA
+ l)r"(l - H 2 ) > n ( / ) ( z ) |
(log
J ^ ) '
log — ^ V x PI /
OO
< ( l - r ) 2 ^ ( n + l)r"K(/)|| l o g B ( Q n=0 <
SUp ||0V,(/) || l o g *}(„,„,. n£ATU{0}
As a result, letting r —> 1, ||/||logB(„,^ <
SUp neNU{0}
|kn(/)||logB(ai/9) < 0 O ,
and then / € logB(Q>/g). Summarizing, 8up{lkn(/)||iogB (aifl) : n e NU {0}} = ||/|| The proof is now complete. We know that Ba = logJ5( a0 )i a n d the relationship between Ba space and Cesaro means is given in [4] as the following corollary. Corollary If / e H(D), then the following statements are equivalent: i) f£Ba,
ae(0,co);
ii) K ( / ) | | B « = 0(1). a
iii) \\tr'n(f)\\«> = 0(n ),
n^w, n-^oo.
References 1. K.R.M. Attele, Toeplitz abd Hankel operators on Bergman spaces, Hokkaido Math. J., 211(1992): 279-293. 2. F. Holland, D. Walsh, Criteria for Membership of Bloch Spaces and its Subspace, BMOA, Math. Ann. 273(1986):317-335, . 3. T. W. Korner, Fourier Analysis, Cambrige University Press, London, 1988. 4. H. Wulan, M. Zhan, Cesaro Means and Complex Function Spaces, Journal of STU, 20(l)(2005):3-6.
I N F I N I T E SERIES IN J A P A N E S E MATHEMATICS OF T H E 18TH CENTURY*
MITSUO MORIMOTO International Christian University 3-10-2 Osawa, Mitaka Tokyo, 181-8585, Japan E-mail: [email protected]
During the Edo period, mathematics was practised in Japan almost independently of European influence. Seki Takakazu and his disciple Takebe Katahiro learned from the Suanxue Qimeng how to express one-variable polynomials with numerical coefficients on the counting board and developed this idea using the so-called side writing method, by which they could treat one-variable polynomials with polynomial coefficients. (See 4 . ) This mathematical tool also helped them t o investigate various properties of the circle. For example, they were interested in representing arc length s in terms of a given diameter d and a sagitta c. Since, in modern notation, s/2 = d arcsin y/c/d, one of their results can be described as follows. Let g\(i) = (1 — \ / l — *)/2 and gn(t) = gi(gn-i(t)) for n = 1,2,3, • • •. Then Takebe recognized lim 4 " s „ ( t ) = (arcsin Vt)2,
(0 < t < 1)
71—+00
Using this formula, Takebe was able to derive the Taylor expansion formula of the inverse trigonometric function (arcsin t ) 2 , which he found earlier by means of numerical calculation. (See 5 .)
Keywords: Katahiro.
Japanese Mathematics, the 18th century, Trigonometric functions, Takebe
2000 Mathematics
Subject Classification:
01A27, 01A50, 33B10
1. Takebe Katahiro Takebe Katahiro (1664 - 1739) was one of the greatest Japanese mathematicians of the Edo Period (1603 - 1868). (There is no standard reference book on the history of Japanese mathematics in English. The most informative book in Japanese is 6 . See also 2 and 8 .) His master Seki Takakazu (c.1640 - 1708) initiated a theory of polynomial algebra with variable coefficients. This theory was called the side writing method and Seki and Takebe applied it to various problems in the Hatsubi 'Abbreviated title "Japanese Mathematics of the 18th Century" or "Infinite Series".
203
204
Sampo Endan Genkai (see 7 ) and later to the study of circles. For the mathematics of Seki Takakazu see his collected work (see 1 ) . The main mathematical works of Takebe Katahiro are as follows: • the Kenki Sanpo (Mathematical Methods to Investigate from Slight Signs, 1683) • the Hatsubi Sanpo Endan Genkai (Colloquial Commentary on Operations in the Hatsubi Sanpo, 1685) — Seki Takakazu: the Hatsubi Sanpo (Methods to Explore Subtle Mathematical Points, 1674) • the Sangaku Keimo Genkai Taisei (Great Colloquial Commentary on the Suanxue Qimeng, 1690) — Zhu Shijie : the Suanxue Qimeng (Introduction to Mathematics, 1299) • the Taisei Sankei (Great Accomplished Mathematical Treatise, 20 volumes, 1710) • the Koritsu (Arc Rate, date unknown) • the Tetsujutsu Sankei (Mathematical Treatise on the Technique of Linkage, 1722), Manuscript of Kokuritsu Kobunsho-kan Naikaku Bunko. • the Fukyu Tetsujutsu (Fukyu's Tetsujutsu, 1722), Manuscript of the University of Tokyo Library. • the Enri Kohaijutsu (Studies on the Circle — Methods to Calculate the Length of Circular Arc, date unknown) • the Sanreki Zakko (Various Considerations on Mathematics and the Calendar, date unknown)
2. Takebe's Infinite Series Expansion In the Tetsujutsu Sankei Takebe Katahiro described the following infinite series expansion of an inverse trigonometric function: Proposition 2.1. [Takebe Katahiro, 1722] (arcsin\/t) 2
- f +
#_ %t?_ &_ 3~+45~+35"
= *d +
5
d +
n
+
d + Ud
128t5 1575
+
+ ^
128t6 1024t7 + 2079 21021 + " '
W
(i + 33-a + 9 1 - a + - ) ) ) ) ) } -
Whereas the first notation is standard in today's mathematics, the second was standard in 18th century Japan. (The second notation requires less multiplications
205
than the first.) The coefficients in the second formula obey the following rules:
1 3
8 15
9 14
32 45
25 33
72 91
1
2-22
32
2-42
52
2-62
2 - 1 2 2 - 2 2 2 - 3 2 2 - 4 2 2 • 5 2 2 • 62 2 - 3 3-5 4 - 7 5-9 6-11 7-13
The first rule which relies on parity is given in the Tetsujutsu Sankei and the second more general rule is given in the Enri Kohaijutsu. Judging from this, the latter book is believed to have appeared later than the former. The manuscript Enri Kohaijutsu has been preserved and handed down by Seki's followers as one of the confidential books of his school.
3. Squared Half Back Arc and Sagitta If the diameter d and the sagitta c are given, Takebe Katahiro and his contemporary Japanese mathematicians can calculate the "squared half back arc" (s/2) 2 to any accuracy by the rule of right triangle (i.e., Pythagorian Theorem) and by repeated application of an acceleration method. In his studies of the circle, Takebe's most important result is to give a formula for (s/2) 2 in terms of d and c, which we will now describe using modern mathematical terminology. (See Figure 1.) Similarity of two right triangles implies that x : c = d : x and x2 = cd. For % the right triangle with hypotenuse d, we have sin# = — = ycld. Since 6 is also a d d s central angle, we have —6 = —. Therefore, the half back arc is given as follows:
Taking the square of both sides, we obtain
I-J
2
/
,
x2
= (darcsin i/c/rfl
206
Figure 1
4. Numerical Method to Find the Infinite Series Expansion Chapter 12 of the Tetsujutsu Sankei starts as follows: In the search of the form and attribute of the back arc, the true number is hidden if it is close to the half circle and the true number appears if it is close to the side. If it is close to the half circle, it belongs to the latitude and its curve is rapid; if it is close to the side, it belongs to the longitude and its curve is slow. Therefore, taking the sagitta to be extremely small, we should search for the number and seek the procedure. Now Takebe Katahiro calculates a numerical value of the "squared half back arc" (s/2) 2 for d = 10 and c = 10~ 5 . To do so, he applies the iterated acceleration method, which he employes for the calculation of the IT to more than 40 digits in Chapter 11 of the Tetsujutsu Sankei. Relying on this numerical value of (s/2) 2 he obtains (1) in Chapter 12 of the Tetsujutsu Sankei. To show a hint how Takebe finds the formula, we reproduce his calculation using the computer algebra system Mathematica as follows: d=10;c=10"{-5}; N[h=cT2(ArcSin[Sqrt [c/d]] )~2,40] {0.0001000000333333511111225396906666728234777}
207 N[c d,40] {0.0001000000000000000000000000000000000000000} N[h-c d,40] {3.333335111112253969066667282347769479596 10~{-11}} N[h-c d -c'2/3,40] {1.777778920635733333949014436146262542024 10"-[-17}} N[8/45,40]
0.1777777777777777777777777777777777777778 Looking at the numerical value of (s/2) 2 , Takebe assumes its principal part is cd. He then finds the first error (s/2) 2 — cd to be approximately equal to 3.33 x 1 0 ~ u , and assumes its principal part to be c 2 /3. He then finds the second error (s/2) 2 — cd — c2/3 to be approximately equal to 1.77 x 10~ 17 , and assumes its principal part to be 8c 3 /(45d). Continueing in this manner, he finally obtains the infinite series expansion formula (1). 5. Approximation Formulas by Interpolation In the Katsuyo Sanpo Seki Takakazu obtained the numerical values of the "back arc" s = 2darcsin y/c/d for
208
By this fractional approximation formula, the approximate values of the "squared back arc" for the given data are as follows: K[c_, d_] : = (17243148700 c d~4 - 27148244837 c~2 d*3 + 11453384892 c~3 d~2 - 807998619 c"4 d - 27148244837 c~2 d~3 - 45408726 c"5)/ (4310787175 d"3 - 8223990414 c d"2 + 4838317774 c"2 d - 845423484 c"3)
N[K[{1, 2, 3 , 4 , 45/10, 5 } , 1 0 ] , 25] {41.40936770200963809032453, 85.98764213357462274474809, \ 134.3928914446184288161436, 187.5361547834501488901178, \ 216.2749378048595665694282, 246.7401100176385622737240} These values are exact up to 8 digits and the graphs of the two functions G(c, 10) and K(c, 10) appear to coincide (Figure 2). P l o t [ { G [ c , 1 0 ] , K[c, 1 0 ] } , -Cc, 0, 5}]
2sa
150 100 SO
1
2
3
4
5
Figure 2 The difference of these two functions is less than 1 . 5 x l 0 ~ 8 a s seen in Figure 3: P l o t [ G [ c , 10] - K[c, 1 0 ] , -Cc, 0, 5}]
209
L.5X10
1x10 5X10
•5X10
Figure 3
6. Algebraic Method to Find the Infinite Series Expansion Later in the Enri Kohaijutsu Takebe introduces an algebraic method to obtain the infinite series expansion formula (1). This method relies on the following proposition:
Proposition 6.1. Put
9i(t)
VT^t
9n{t) = gi(g„-i(t)),
n= 1,2,3, •
Then we have
lim 4ngn(t) = (arcsin V~t)2,
(0 < t < 1).
(2)
From Figure 4, the relation x : c = d : x implies x2 = cd. By the rule of right triangle, x2 + y2 = d2, and hence y2 = d2 — cd. It follows that the sagitta c' of the half back arc satisfies d/2 — c' = y/2 and is given by c' = (d— %/d2 — cd)/2. We call c' the A-sagitta. Letting g(t) = (1 — y/1 — t)/2, we have c'/d = g(c/d).
210
Figure 4 Since the first approximation of (s/2) 2 is cd and that of (s/4)2 is c'd, the second approximation of (s/2) 2 is given by 4c'd = 4d2g(c/d). We call c' the A-sagitta. Since the second approximation of (s/4) 2 is 4c" d = 4d2g{c'/d) =
4d2g(g(c/d)),
the third approximation of (s/2) 2 is given by 4 2 c"d = 42d2g{c'/d) =
42d2g(g(c/d)).
We call c" the B-sagitta. Iterating this argument, we find l i m , , - ^ 4nd2gn(c/d) geometrically.
= (s/2) 2 , which proves (2)
7. Counting Board Algebra and Generalized Division There was no notion of the Cartesian plane in traditional Japanese mathematics, consequently no notion of the graph of a function nor notion of the tangent. This means that there was no notion of differentiation as we have in real variable calculus. However, Takebe Katahiro and other Japanese mathematicians of the 18th century learned the manipulation of polynomials from Chinese mathematics (The Suanxue Qimeng (Sangaku Keimo in Japanese), Zhu Shijie, 1299). (For the history of Chinese mathematics, see, for example, 3 .) Let P{x) = a + bx + ex2
211
be a polynomial with numerical coefficients and q be a number. Japanese mathematicians could calculate freely the coefficients a', b', d of P(x) =a' + b'(x -q)+
c'{x - q)2
from the given values a, b, c, and q. In today's terminology, this is nothing but the Taylor expansion of the given polynomial
P"(a\ P(x) = P(q) + P'(q)(x -q) + t-M{x
-
qf.
In this sense they knew the derived polynomial P'(x) and the fact that the polynomial function P(x) takes extreme values at x = q when P'(q) = 0. (See Chapter 6 of the Tetsujutsu Sankei.) In traditional Chinese and Japanese mathematics, a polynomial with numerical coefficents was represented by the column vector whose components are the coefficients of the polynomial. The polynomial P(x) was represented on a counting board using counting rods as follows: Quotient row a Reality row b Square row c Side row Corner row
shang shi fang lian yu
1 Quotient row a' Reality row b' Square row c> Side row Corner row
shang shi fang lian yu
In the sequel, we turn the column vector 90 degree clockwisely and represent it as a row vector. The calculation of a', b' and c' from the old coefficients a, b, c and the number q is done by adding upward from the bottom to the top several times (in the row vector notation, from the right to the left). This algorithm is called generalized division and can be traced back to the Jiuzhang Suanshu (Nine Chapters of the Arithmetic Arc) of the Chinese Han Dynasty. a b c a b c 1) (b + cq)q cq a + (b + cq)q b + cq c cq a + (b + cq)q b + 2cq c Q To calculate a", b", c" we add upward two times:
P(x) = a + bx + cx2 = a' + b'(x-q)+c'(x-q)2 = a" + b"(x -q-q') + c"(xq - q')2
212 b b cq b + cq cq q a + (b + cq)q b + 2cq b' q a' b< q + W) a.' c'q' (&' + c'q')q' a' + (&' + c'q')q' U + c'q' (q)
a a (6 + cq)q a + (b + cq)q
c c c c d c' d
C'q'
q + q' a' + (&' + c'q')q' b' + 2c'q' d b" c" q + q' a" If the Reality row becomes null after several operations, the number at the Quotient row q + q' + • •• gives a solution of the equation P(x) = 0. If P(x) = x2 — 2, this calculation is called the extraction of the square root; if P(x) = x3 — 2, this is called the extraction of the cubic root. It was in these forms that this algorithm was introduced in the Jiuzhang Suanshu. If P(x) = a + bx, this algorithm is the usual algorithm of division to find the decimal expansion of — b/a. Therefore, this algorithm is called generalized division. 8. The Taylor Expansion Formula in Japanese Mathematics 8.1. Case of a Quadratic
Equation
Note that x = g(t) = (1 — \ / l — t)/2 is a solution of the equation P(x) = -t + 4x-
4x 2 = 0.
If x and t are small, neglecting the terms of higher degree, Takebe finds the equation P{x) = 0 is approximated by —Ax + t == 0. Therefore, he expands the equation with respect the new variable x\ — x — - . The calculation is done by generalized division. We can recalculate it using the computer algebra system Mathematica as follows: P[x_] = 4x"2 - 4x + t P l [ x l _ ] := P[xl + t / 4 ] Expand [PI [ x l ] ]
Pi{Xl)
t t2 = P(Xl + - ) = - - + (4 - 2 t ) n - 4x? = 0.
Neglecting the terms of higher degree in x\ and t, Takebe finds the equation t2 Pi(xi) = 0 is approximated by — — +4:ci =F 0. Therefore, he expands the equation
213 t2 t with respect to the new variable x2 = x\ — - — = x — 4-4 4 division, he finds P 2 (z 2 ) = Pi(x2 +1 2 /16) = - j
t2 - . Using generalized 16
- ~ + (4 - 2t - t-)x2 - 4x22 = 0.
Neglecting the terms of higher degree in x2 and t, Takebe finds the equation t3 ^2(^2) = 0 is approximated by — — +4x2 4= 0. Therefore, he expands the equation 8 i3 with respect to the new variable x3 = x2 — -—-. Using generalized division, he 8•4 finds
p3(x3)=p2(x3+y 5<4 t5 t6 = -64-64-256
,t + ( 4
-
n 2 t
t2 t\ -¥-4)a:
o 3
-
4 a ; 3
n
=
a
Neglecting the terms of higher degree in x3 and t, Takebe finds the equation 5i 4 P3(x3) = 0 is approximated by — — +4^3 4 0. Therefore, he expands the equation 64 5t 4 with respect to the new variable Xi = x3 — ——-. Using generalized division, he 64 • 4 finds 5£4 P 4 (x 4 ) = P3(x4 + — ) 7t5 128
7t6 5i 7 25t8 512 1024 16384 t2 t3 5t 4 + ( 4 - 2 * - - - - - — )x* ~ 4x1 = 0. Neglecting the terms of higher degree in x± and t, Takebe finds the equation 7t4 Pi(x4) = 0 is approximated by — TT^T+4X4 = 0. Therefore, he expands the equation 128 7£5 with respect to the new variable x& = X4 — —-—-. Using generalized division, he 128 • 4 finds P5(x5)=P,(x5
+
7t5 —)
21t6 3t7 81t 8 35t 9 49t10 _ 512 ~ 256 16384 ~ 16384 ~ 65536 /, „ t2 t3 5*4 7t s . , , By this repeated application of generalized division, he can find any number of terms in the Taylor expansion of the A-sagitta g(t): , x * t2 t3 5t4 It5 + + + + > = 4 16 32- 2.56 5T2- + - - - '
s(i
(3)
214 where the coefficients are called, from the lowest order term, the A-original number, the 1st A-difference, the 2nd A-difference, the 3rd A-difFerence and the 4th Adifference, etc. 8.2.
Iteration
Since -t + 4g(t) -4g(t)2 = 0, we have -g{t) + 4g(g(t)) -4g(g(t))2 x from the system of equations -t + 4x - 4x2 = 0,
= 0. Eliminating
-x + 4y - 4y2 = 0,
we find that g(g(t)) satisfies —t + 4(4y — 4y2) — 4(4?/ — 4y2)2 = 0. Therefore, if we employ the equation of degree 4 -t + 16j/ - 80j/2 + I28y3 - 64j/4 = 0, we can expand the function g(g(t)) by repeated application of generalized division. The iteration in the Enri Kohaijutsu is simpler: Takebe uses the quadratic equation Q(y) = -9(t) + 4y-4y2
=0
and the expansion formula (3). Namely, put ^. , . t t2 t3 ^ 4 16 32 Neglecting the terms of higher Q(y) = 0 is approximated by —1/4 v w
5t4 7t5 21i 6 „ „ 2 „ 256 512 2048 order in y and t, Takebe finds the equation + 4y = 0. Therefore, he expands the equation
with respect to the new variable y\ = y — -—T. Using generalized division, he finds Qi(yi) = Q(yi + 5i =
2
^ ) t
3
5i 4
7t 5
21t 6
,A h (4
t.
A 2
n
)j/i - 4yf = 0.
yl 64 32 256 512 2048 v 2,y Neglecting the terms of higher 2 order in jyi and t, Takebe finds the equation bt Qi(yi) — 0 is approximated by -+4yi == 0. Therefore, he expands the equation 5t 2 with respect to the new varianble 2/2 = Wi — TT.—:• Using generalized division, he 64 • 4 finds 5t2 Q 2 (2/2)=Ql(2/2 + 6 4 - 4 ' 21i 3 345t4 7i 5 21t6 .„ t 5t2. „ A 2 = h v(4 )y2 - 4u,2 = 0. ' 512 16384 512 2048 2 32 J y " Neglecting the terms of higher order in j / 2 and t, Takebe finds the equation 21£3 (32(2/2) = 0 is approximated by ——— + 4y2 = 0. Therefore, he expands the 512
215 21t3 equation with respect to the new variable 3/3 = 2/2 — irrz—r- Using generalized 512 • 4 division, he finds 21i 3 Q3(2/3) = Q2(ys + 5 1 2 4 ) 429i 4 lOOlt5 11193t6 .„ t 5t 2 21i 3 . „2 „ h vf4 )?/•) - 4vi ya = 0. ' 16384 65536 1048576 2 32 256 'y Neglecting the terms of higher order in 2/3 and t, Takebe finds the equation 429£4 QiitJi) = 0 was approximated by — + 42/3 = 0. Therefore, he expands the 16384 429£4 equation with respect to the new variable 2/4 = 2/3 ~ _ .—-. Using generalized 16384•4 division, he finds 429£4 Q4(2/4) = <33(2/4 + 16384-4' 2431i5 24531*6 9009£7 18404K8 • 131072 2097152 16777216 1073741824 ,„ t 5t2 21t 3 429£ 4 , , 2 „ + ( 4 - 2 - 3 2 - - 2 5 6 - 8 1 9 2 ) 2 / 4 - 4 2 / 4 = °Neglecting the terms of higher order in 2/4 and t, Takebe finds the equation 2431t5 (34(2/4) = 0 is approximated by — i c . l r . „ 0 + 4j/4 = 0. Therefore, he expands the =
equation with respect to the new variable 2/5 = 2/4division, he finds ^
/
N
^
2 4 3 1
1
*
5
243 It5
. Using generalized
>
<3s(2/5) = <34(2/5 + 1 3 1 0 7 2 .4) 29393i6 529147 592449£8 1042899*9 5909761*10 ' 2097152 4194304 1073741824 4294967296 68719476736 t 5t2 21i 3 429£4 24311\ , 2, „ + v(4 )y5 - 4um = 0. 2 32 256 8192 65536 Jyb By this repeated application of generalized division, he can find any number of terms in the Taylor expansion of the B-sagitta 2/2(t) = g(g(t))', , „ i/U/V ))
5t 2
t
n(q(t)) =
1 1 6
T-
21t 3
429£4
1 25g
-r
1 2 0 4 g
-r
1 6 5 5 3 g
,
-r
where the coefficients are called the B-original number, the 1st B-difference, the 2nd B-difference, the 3rd B-difference and the 4th B-difference, etc. 9. Taking the Limit 9.1. Further
sagittas
Taking the diameter d = 1 and the sagitta c = t, the A-sagitta, B-sagitta, etc. can be given as follows:
216 A-sagitta B-sagitta C-sagitta D-sagitta E-sagitta F-sagitta G-sagitta H-sagitta I-sagitta J-sagitta
gi{t) g2{t) g3(t) g4(t) g5{t) g6(t) g7(t) ga{t) g9(t) flio(t)
= g(t) = g(g(t)) = g{g(g{t))) = g{g{g{g(t)))) = g(g{g{g(g{t))))) = g(g{g(g(g(g(t)))))) =
In the Enri Kohaijutsu, Takebe does not apply generalized division to obtain the C- and further sagittas. First, he writes down the rule by which he obtains the expansion of the Bsagitta from that of the A-sagitta. For example, The B-original number is obtained by dividing the A-original number by 4. The 1st B-sagitta is obtained from the A-original number and the 1st A-difference. The 2nd B-difference is obtained by the A-original number, the 1st A-difference, and the 2nd A-difference, etc. Second, he applies this rule to the expansion of the B-sagitta to obtain the expansion of the C-sagitta. Using Mathematica, we reproduce his calculation of the expansion up to the term of tn (10 terms) for the A-sagitta, the B-sagitta, the C-sagitta, • • •, the J-sagitta. .
t2
, ,
4SlW = t + T ,
2
N
w u
, 44
,x yJV ; 94[t) -t+
5 95
W ~
l +
i3 +
5t 2 16 21t2 64 ^f.
5t4
- +-
7t5 +
-
+
...
21t 3 429t4 2431t5 128 4096 32768 357t 3 29325i4 666655t5 2048 262144 8388608 5797t3 1907213t4 173556383t5 + + + + 2147483648 '" 256 3276g 16777216 2 3 4 341t 93093t 122550285t 44616473759t5 1024 + 524288 + 1073741824 + 549755813888 + " '
9.2. Taking the Limit
Numerically
In the Enri Kohaijutsu Takebe calculates numerically the following coefficients in the above expansion formulas: N [ { l / 4 , 5/16, 21/64, 85/256, 341/1024}] {0.25, 0.3125, 0.328125, 0.332031, 0.333008} From this, we find that the coefficient of t2 in the limit f(t) is equal to - . o Next, Takebe calculates numerically the ratio of coefficients:
217 N[{1/8/(1/4), 21/128/(5/16), 357/2048/(21/64), 5797/ 32768/(85/256), 93093/524288/(341/1024)}] { 0 . 5 , 0.525, 0.53125, 0.532813, 0.533203} N[8/15] 0.533333 1 8 is equal to - x — . 3 15 Continuing in this manner, Takebe gives a different proof to t h e infinite series
From this, he finds t h a t t h e coefficient of t3 in t h e limit fit) expansion formula (1). 1 0 . T h e Sanreki
Zakko
T h e manuscript of t h e Sanreki Zakko was held a t t h e "Shokokan" in Mito city. T h e "Shokokan" was t h e library of one branch of t h e Tokugawa family b u t was destroyed by fire during t h e 2nd World W a r in 1945. Fortunately, t h e manuscript was copied chemically in 1944 a n d reproduced recently in 9 . In this book several approximation fractional formula and t h e addition formula for t h e trigonometric functions sin t a n d cos t were introduced. Using these formulas, t h e table for sin was constructed. This is t h e first occurrence of t h e trigonometric function in J a p a n e s e m a t h e m a t i c s . References 1. Hirayama, Akira; Shimodaira, Kazuo and Hirose, Hideo (ed.), Seki Takakazu Zenshu (The Collected Works of Seki Takakazu, in Japanese with English summary), Osaka Kyoiku Tosho, 1974. 2. A. Horiuchi, Les Mathematiques Japonaises a I'Bpoque d'Edo, Vrin, 1994. 3. H. C. Martzloff, A History of Chinese Mathematics, Springer 1997. [the French original edition: Histoire des mathematiques chinoises, Masson, Paris 1987.] 4. M. Morimoto, A Chinese root of Japanese traditional mathematics, Proceedings of the 11th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications, Wasan, (2003):133-141. 5. Morimoto, Mitsuo and Ogawa, Tsukane, Takebe Katahiro no Sugaku — Tokuni Gyakusannkaku Kansu ni kansuru Mittsu no Koshiki ni tsuite — (The Mathematics of Takebe Katahiro — His Three Formulas for an Inverse Trigonometric Function, in Japanese), Sugaku, 56(2004):308 - 319. (English translation will be published by AMS.) 6. Nihon Gakushiin, Nihon Kagakushi Kankokai (ed.): Meiji-zen Nihon Sugakushi (History of Japanese Mathematics before the Meiji Restoration, in Japanese), Shinteiban (New Edition)", in 5 volumes, Noma Kagaku Igaku Kenkyu Shiryo kan, 1979. (First edition, Iwanani Shoten, 1954 - 1960.) 7. Ogawa, Tsukane: Seki Takakazu Hatsubi Sanpo, Gendaigoyaku to Kaisetsu (Seki Takakazu's Hatsubi Sanpo, Its translation to modern Japanese with annotation, in Japanese), Osorasha, 1994. 8. Ogawa, T.: A review of the history of Japanese mathematics, Revue d'histoire des mathematiques, 7(2001):101-119. 9. Sato Ken'ichi, Sr.: Takebe Katahiro no Snreki Zakko — Nihon hatsu no sankaku kansu hyo (Takebe Katahiro's Sanreki Zakko — Table of trigonometric functions first in Japan, in Japanese), Kenseisha, 1995.
M E R O M O R P H I C F U N C T I O N S THAT S H A R E F O U R SMALL F U N C T I O N S
QIU GANGDI Department of Mathematics Ningde Teachers College Ningde, Fujian 352100 P. R. China E-mail: [email protected]
In this paper, we mainly discuss the uniqueness problem of k order derivatives of meromorphic functions that share four small functions, and consider Nevanlinna's Four Values Theorem when the derivatives of meromorphic functions share some small functions, which generalize and improve some related results of the author.
Keywords: Meromorphic function, small function, uniqueness. 1991 Mathematics
Subject Classification:
30D35
1. Introduction and Main Results In this paper, we use the same symbols as given in Nevanlinna theory of meromorphic functions (see 1 ) . By S(r, / ) we denote any quantity satisfying S(r, / ) = o{T(r, / ) } as r —> oo, possibly outside a set finite linear measure. A meromorphic furction a is said to be a small function of / if and only if
T(r,a) = S(r,f) Let f(z) and g(z) be two nonconstant meromorphic functions, and a(z) a small function of f(z) and g(z), if f(z) — a(z) and g(z) — a(z) have the same zeros ignoring (counting) multiplicities, then we say that f(z) and g{z) shares a(z)IM(CM). In addition, let S(f = a = g) be the set of all common zeros of / — a and g - a, S(ktTn)(f = g = a) be the set of all points which are zeros of / — a with multiplicities k as well as the zeros of g — a with multiplicities m. Funthermore, we denote SE(f = a — g) the set of all common zeros of / — a and g — a with the same multiplicities. Denote N(r, f = a = g), 218
219
N(k,m) (r> / = a = 9) a n < i NE ( r ' / = a = d) t n e re duced counting functions of / and g correspondent to the sets S(f = a — g), S^^ (/ = 9 = a) and SE(f = a — g), respectively. _ If N(r, l / ( / - a)) - W(r, f = a = g) = S(r, / ) , and N(r, l/(g - a)) N(r, f = a = g) = S(r, g), then we say that / and g share a IM*. _ If N(r, l/(g - a)) -NE(r,f = a = g) = S(r, / ) , and N(r, l/(g - a)) NE(r, f = a — g) = S(r,g), then we say that / and g share a CM*. In 1929, R. Nevanlinna proved the following famous Four Values Theorem: Theorem A (see 2 ): Let / and g be two nonconstant functions, aj(j = 1,2,3,4) be four distinct complex numbers. If / and g share aj (j = 1,2,3,4) CM, then / is a Mobius transformation of g. In 1998, the author proved the following theorem when considering derivatives of meromorphic functions that share four small entire functions: Theorem B (see 3 ) Let / and g be two nonconstant meromorphic functions, k be a positive integer, and a,j(j = 1,2,3,4) four distinct small entire functions with respect to / and g. If f^ and gW share bj(j = 1,2,3,4) CM*, then/ ( f c ) = g{k). In this paper, we generalize and improve the result of theorem B and obtain the following result: Theorem 1.1. Let f and g be two nonconstant meromorphic functions, k be a positive integer, and a\, ai, 03 three distinct small functions with respect to f andg. If f^ andg^ share 00, a\, a^lM*, and share a$CM*. then one of the following cases does occur: (i) fW = g(k) (ii) (/(fe) — ai)(g(k^ — ai) = (ai — CI2)2, and 00, a\ are two exceptional functions of f^ such that ai + 03 = 2a\ (Hi) (/(fc) — ai)(g^ — 02) = (0-2 — os)2, and 00, ai are two exceptional functions of f^ such that a\ + 03 = 2a2 (iv) (/( fc ' — az){g(k> — az) = (03 — a\)2, and 00, as are two exceptional functions of f^ such that a\ + 02 = 203 2. Some Lemmas Lemma 2.1. Assume the condition same as given in theorem 1.1, then S(r,fW)
= S(r,gW)±S(r)
(1)
Proof. It is easy to see that (1) follows by the conditions of theorem 1 and Nevanlinna's three small functions theorem (P. 47 1 ).D
220
Lemma 2.2. (see 4>5) Let f be a nonconstant meromorphic function, b\ and 62 be two distinct small functions of f, and / - 61 6 1 - 6 2
L{fMM)
f'-K
&i-&2
(2)
then («) L ( / A , 6 2 ) = 0
£
(ii)
m[r>
(3)
( / ^ » y ] = S(r,f),(i = l,2)
(4)
(5)
/• N L(f,bi,b2) , c , ,.>. r (w) "i[r, , , , w , , J = S(r, / ) (/-&1K/-62) where a is a small function of f 2
(6)
2
<"> 5 X ' ' 7/ ~ ^& >i - ^ i=l
7L(f,h,b 7 7 2^)') * £ * < ' . T^T) / - 6 i + 5(r, /) (7) t=i
6
Lemma 2.3. ('see J Let / and 5 fee two nonconstant meromorphic functions, aj (j = 1, 2, 5, ^J 6e /our distinct small functions with respect to f and g. If f and g share a\, a2CM*, and share 0,3, a^IM*, then ag + b eg + d
(8)
where a, b, c, d are small functions with respect to f and g. 3. Proof of Theorem Let JP f(k) ~ a i n 9(k) - «i 03 - ffli r = , Cr — ,a — a
(9)
(10)
221
from (4)-(6) of Lemma 2.2 we deduce that m(r,H) = S{r)
(11)
It is easy to see that the poles of H only possibly comes from the zeros, poles, 1-points and a-points of F. Combining (7) with (10), we can know that 1 and a are not the poles of H. Next, (10) can be rewritten as
;(^r-f)(fe-#)(fe-f)](^-G)
H
(12) F G -a F-l G-a
G G-
therefore, 0 is also not a pole of H. In addition, from (10) we get H =
F-G FG
F'(aG'-a'G) (F-l){G-a)
G' jaF' - a F) (G-l)(F-a)
By a simple computation we have H = F-G FG
~a(a - l)F'G' + a'FG'{F - 1)(G - a) - a GF'{G - 1)(F - a)
(F-l)(F-a)(G-l)(G-a)
A R ~ Q
(13) Now let ZQ is a pole of / with multiplicities m as well as a pole of g with multiplicities n, from (9) we know that ZQ is a pole of F with multiplicities 77i + /c as well as a pole of G with multiplicities n + k. Then it follows that ZQ is a pole of R with multiplicities 2m + 2n + 5k + l+max(m, n) at most, as well as a pole of Q with multiplicities 3(m+n+2k) Noting that: 3(m + n + 2k) — [2m + 2n + 5k + 1 + max(m, n)] = m + n + k — 1 — max(m, n) ^ 1 So ZQ is not a pole of Q but a zero of H. According to the discussions above, we get N(r,H)
= S(r),
T(r,H)
= S(r).
which leads to
222
Hence N(r, F) ^ N(r, £) < T(r, H) + O(l) = S(r)
(14)
Therefore F and G snare 0, I, IM*, and share oo, aCM*. From lemma 2.3 we get
F=
°-4±^
c3G + c4
(15) v
;
where ci, c-i, Cz, C4 are small functions with respect to F and G. If none of 0, 1, a is the Picard exceptional function of F and G, substituting sharing small functions 0, 1 and a of F and G into the (15), then we deduce that F = G, and /(fc) = g^ according to (10), which is a contradiction. Hence, one of 0, 1 and a must be the Picard exceptional function of F and G. From (14) we known that oo is a Picard expectional function of F and G, it follows that among 0, 1 and a, there is one and just only one function being an exceptional function of F and G. We discuss the following three cases: Case I: 0 and oo are two exceptional functions of F and G. Substituting the common 1-points and a-points of F and G in to (15), we have ct + c2 = c3 + ci,cla + c2 = c3a2 + c4a
(16)
Next, from (15) we get
c3G + c^
(17)
So, jj2- = 0, or oo, and it follows that c2 = 0, or c, = 0 (1.1) If c2 = 0 , substituting it into (16), we deduce that a = 1, which is a contraction. (1.2) If Cj = 0, combining (16) with (17),we have Co
c3G + c4
,c4 = —c3(a + 1) and c2 = —c3a.
Hence G-{a+l)
(18)
which leads to a + 1 = 0, i.e, a = — 1 From (18) we obtain that FG = 1, together with (9) we get (/ (fc) a1)(g(-k^ — a^ = (a2 — ax)2,a2 + a3 = 2a15 and oo, aa are two exceptional functions of /(fe) and g ^ .
223
Case II: 1 and oo are two exceptional functions of F and G, substituting the common zeros and a-points of F and G into (15) we get c2 = 0, cx = c3a + c4
(19)
From (15), we have F=
,C'G.x
(20)
(G+l)
Hence, — ^ = 1 or oo, and it leads to c4 = —c3 or c3 = 0 If c3 = 0, then (19) and (20) leads to F = G, a contradiction. If c3 = —c4, combining (19) with (20) we get (21) Considering that
therefore, j ^ = 1 or oo, it result in a = 2. Substituting it into (22) we have (F-1)(G-1) = 1 Combining (19) with (23) we get (/
(1 - a)G G-a
(24)
(25)
Considering that F_a=(l-2a)G
+
a
G—a 2
1
Therefore, ^frj = a or oo, which result in a = ^ , Combining (9) with (26), we have (/ (fc) -a 3 )(# ( f c ) - a 3 ) = (a3 - a x ) 2 , ax +a3 = 2a 3 , and a3 and oo are two exceptional functions of f^ and g^k\ Which completes the proof of theorem 1.1.
224
References 1. W. K. Hayman, Meromorhic Functions, Clarendon Press, Oxford, 1964. 2. R. Nevanlinna, Le theoreme de Picard-Borel eet la theorie Des Fanctions Meromorphes, Paris, 1929. 3. G. D. Qiu, Four Values Theorem of Derivatives of Meromorphic Functions, Systems Science and Mathematical Sciences, ll(1998):245-248. 4. G. D. Qiu, Uniqueness of Entire Functions that Share Some Small Functions, Kodai Math, J. 23(2003):1-11. 5. Y. H. Li, Entire functions sharing four small functions IM, Acta Math Sinica, 41(1998):249-260(in Chinese). 6. P. Li, Meromorphic Functions that Share Four Small Functions, Math. Anal. Appl. 263(2002):36-42.
I N T E G R A T I O N OPERATORS O N T H E B M O A T Y P E SPACES*
DANQU Department of Mathematics Shantou University Shantou, Guangdong, 515063 P. R. E-mail: [email protected]
China
Let (f> be an analytic function on the open unit disk in the complex plane. We study the boundedness of the following operators Jo a
on Bloch-type spaces B
Keywords:
Jo and BMOA-type spaces.
Ba, BMOAa,
2000 Mathematics
BMOAa,
integration operator, boundedness.
Subject Classification:
47B33
1. Introduction Let D — {z € C : \z\ < 1} denote the open unit disk in the complex plane C and 3D = {z € C : \z\ = 1}. For a € D, 0 < r < 1, let D(a, r) = {w € D : (3(z, w) < r} denote the Bergman disk, where /?(z, w) = 1 loo- l + \
a-z 1-a.z-
Let a > 0, the a — Bloch space Ba is defined to be the space of analytic functions f on D (denoted by H(D)) such that H/IIB-
:= sup(l - |z| 2 ) Q |/'(*)| < +oo, zeD
Note that when a = 1, Ba is the well-known Bloch space B. The space ' T h i s work is supported by NSF of Guangdong Province (Grant No. 032028). 225
226
BMOAa
is denned to be the space of / <E H(D) with ••= sup / (1 - |2| 2 ) 2 Q - 2 |/'(*)| 2 (1 - \
MWBMOA-
a£D JD
The space BMOAa \\f\\BMOAa
is the space of / £ H(D) such that
••= SU P LL— ICdD Ml
where / is any arc | I | , z / | z | e I } , and BMOA1 = BMOAx For <j> £ H(D),
| / ' ( z ) | 2 ( l - \z\2)dA(z)
/
< +0O,
Js(I)
on the unit circle dD, S(I) = {z € D : \z\ > 1 — |/| is the normalized arc length on dD. Note that = BMOA. the operators Js, Is are defined respectively by:
Mf)(*) = f /(O0'(Ode,
h(f)(*) = f /'(O0(Ode, / e #(£>).
Jo Jo The operator J,/, is called the Cesaro operator, and Is is often called the companion operator of Js- The boundedness of Js, Is on analytic function spaces has been studied extensively, see, for example, [l]-[4]. In this article we study the boundedness of the operators Js and Is from Ba (BMOAa, BMOAa) to B13, and the compactness of Is from Ba to B0. Throughout the article, C denotes positive constant whose value is not necessarily the same at each occurrence. 2. The boundedness and compactness of 1$ on B Q Lemma 2.1. ([4]) Let a, (3 > 0, <j) € H{D), the operator Is : Ba -> B0 is compact if and only if for any bounded sequence {fj} in Ba, which is uniformly convergent to zero on compact subsets of D, {fj} satisfies IIM/i)llB<»->0
O'-OO).
Lemma 2.2. ([5]) Fixed a £ D, for z & D(a,r), 1 — \a\2, and comparable to |1 — az\.
1 — \z\2 is comparable to
Theorem 2.1. Let a,(3 > 0 and <j> € H{D). bounded from Ba to B® if and only if
Then the operator Is is
sup(l - |z|y- a |<£(.z)| < 00. zeD Proof. Suppose that C = sup z e £ ) (l - |2;|2)/3_a|(^(2;)| < 00. Then we have IIW)(*)IIB*
= sup(l - \z\2Y\f'{z)4>{z)\ z£D
<
CII/HB-
227
To prove the converse, suppose 1$ is bounded from Ba to B@. Firstly, we prove the case a = 1. For a G D, let fa(z) = log 1 _ 1 az , then it is clear that fa{z) is a bounded set in B. So by Lemma 3 and using the subharmonicity of \<j){z)\ and the fact that fD,ar\ H-M^SdA(z) < C < oo (C depending only on r), we have la^l-lal2)^-1)!^)!2
< irrm I I 1 - \a\ )
(1
JD(a,r)
( 1 - \z\*)W-*>
JD(a,r)
= C [
~ M 2 ) 2 ^w*)l a «w(«) -J°)L—Mz)\*dA(z) \L — aZ\
(1 - | z | 2 ) 2 ^ 2 | ( l o g - ^ ) ' | 2 | « ^ ) | 2 c L 4 ( z )
JD(a,r)
I — az
< C s u p ( l - \z\^\fa{z)\2\
/ JD(a,r)
< C\\hfa\\le I JD{a,r)
) U
-
\z\
,2.2dA(z) )
—J—dA(z) I 1 - \z\
)
< C\\I,f,\\2\\fa\\2B < <X>.
In the case a ^ 1, by putting fa(z) = (1 — az)l~a, we can prove that as well, so we omit it. Remark: (1) R. Yoneda showed that if a > 1, then 1$ is bounded on Ba if and only if 0 € H°°([S\); if /3 > a > 1, then 1$ : Ba -> B? is bounded if and only if sup, e I 3 (l - \z\2)a-^\<j>'{z)\ < oo([2]). (2) The boundedness of 1^ : Ba -> B' 3 can also be obtained by the boundedness of Jj, : Ba -> B' 3 and M^ : Ba -> B^{M
Then 1$: Ba -> B " is compact
lim(l-|z|Y-«|^)|=0.
|z|-»i
Proof. Suppose that l i m i ^ ^ l - |z| 2 )^ - a |^(2;)| = 0, then for any e > 0, there exists 5 e (0,1) such that (1 - \z\2)^-a\(p(z) < e as \z\ > 1 - 5. If {fj} is bounded in Ba, and {/,} is uniformly convergent to zero on compact subsets of D, {/•} uniformly convergent to zero on compact
228
subsets of D, then for the e > 0 above, there exists N £ N such that \f'j(z)\ < e for all z £ {w : \w\ < 1 - 6} as j > N. Thus
IIVilla* <
sup ( l - | z | W ; ( z ) | W z ) | +
sup ( l - | z | Y | / j ( z ) | | ^ ) |
|z|
\z\>l-6
sup ( l - | z | y - a | ^ ) | | | / , - | | B -
|z|>l-<5
< ( C + ||/il|fl-)e, when j > TV, where C = s u p z e D ( l — |z|2)/3|i?!>(z)| < oo. Conversely, suppose lim| z |_i(l — |z|2)/3_Q|(/>(2:)| ^ 0, then there exists e 0 > 0 and {ZJ} c D, \ZJ\ -> 1 (j -> oo) such that (l-|^| 2 ) 0 - a |(zj)l > £oi
I
. 12
Let fj(z) = n-z'z)" • ^ *s e a s y *° c n e c k that fj £ 5 a is {fj} convergent to zero on compacts subsets of D, we have
uniformly
II V I I B * > (1 - N Y l / j ^ M ^ ) ! > eo(l - l ^ | 2 ) Q | / j f e ) l > eo/2. By Lemma 1 this contradicts to the compactness of 1$. 3. The boundedness of J<j, and J<^ on Lemma 3.1. For a>0,
C Ba.
BMOAa
Proof. Suppose / £ BMOAa, [3] give
BMOAa
Lemma 2, the subhamonicity of \f'{z)\ and
(1 - |a| 2 ) 2 "|/'(a)| 2 < ( , 1 " I ^ 2 \ 2 2 Q / (1 - \a\zy JD{a,r)
\f'{z)\2dA{z)
< (1 - lal 2 ) 2 ^- 1 ) /
|/'(2)| 2 (1 - \<j>a{z)\2)dA{z)
JD{a,r)
\f'(z)\2(l-\ct>a(z)\2)dA(z)
I JD
^CsupCl-lal2)2^-1) / a£D
« C sup ^ ICdD
|/'(z)|2(l-|0a(z)|2)dA(z)
JD
^ \-L\
/
|/'(z)| 2 (l - |z| 2 K4(z) < oo,
JS(I)
where the equivalence above is a result in [1]. That is <j> £ Ba. Hence BMOAa C Ba.
229
-> B13 is bounded if and
Theorem 3 . 1 . Let a, (3 > 0. Then Jj,: BMOAa only if
sup z G L ) (l - \z\2)0\(j>'(z)\ < +co, 0 < a < 1; s u P z e D ( l - | z | Y l o g T - | ^ | 0 ' ( ^ ) | < +oo, a = 1; ^ s u p z 6 D ( l - \z\2f-a+x\ct>'{z)\ < +oo, a > 1. Proof. By Lemma 3, BMOAa C Ba, the necessity can be obtained from the boundedness of J4,: Ba -> S' 3 ([4]). Suppose Jtf,: BMOAa —> B13 is bounded. When a > 1, it is easy to see that {fa(z) = (1 - az)l-a) is a bounded set in BMOAa([3}). Using the subharmonicity of |'(z)| and Lemma 4 yield (l-|a|2)2/3-2Q+V(a)|2 (1 - | a | 2 ) ^ - 2 Q + 2 / < 2 2
(i-H )
„
\4>'(z)fdA(z)
JD(a,r)
/• (l-N2)2^!-^2-2"^^)!2.^, JD(a,r) U_ H )
< SUP(1 - |^| 2 ) 2 / 3 |/a(^)| 2 |0'W| 2 / zED
j—^dA{z)
JD(a,r)
t 1 - \z\
)
= CWJ^faWgo < C\\J42\\fa\\BMOAa
where C 2
=
jD{a>r)
< «>,
{1_fzliy2dA(z)
<
+00.
Thus sup z e £ ) (l
-
\A f~
a+1
Wiz)\ <+°°-
When a = 1, by putting fa(z) = log jz^ G BMOA, we can prove that as well as the case of a > 1, so we omit the detail. When 0 < a < 1, since a non-zero constant C belongs to BMOAa, we have J^C € B@. Hence s u p z e D ( l - | z | 2 W ( * ) l < +°oTheorem 3.2. Let a, (3 > 0. Then 1$: BMOAa
—> B13 is bounded if and only if
sup(l - \z\2f~a\(j>{z)\ < +00. z£D
Proof. Using BMOAa C Ba, and choose the same test functions as the function as above, the proof is similar to that of Theorem 1.
230
4. T h e b o u n d e d n e s s of J^ a n d 1$ o n
BMOAa
T h e o r e m 4 . 1 . Let a,/3 > 0. Then J^: BMOAa only if ' s u p z e i ) ( l - \z\Y\4>'(z)\
-> B? is bounded if and
< +oo,
0 < a < 1;
< s u p ^ t l - l z p y ' l o g j ^ p l ^ z ) ! < + o o , a = l; , supz€l,(l - | z | y - a + 1 | 4 > ' ( z ) l < +oo,
a > 1. 3
T h e o r e m 4 . 2 . l e t a, (3 > 0, J^; S M O A " —> B ' is bounded if and only if snp(l
- \z\2)0-a\
Note t h a t BMOAa C B Q , the proofs of Theorems 5 and 6 are similar to those of Theorems 3 and 4 respectively. T h e details are omitted. R e m a r k : In [3], R. Yoneda showed the boundedness of operators J^,,/^,: BMOAa -> B0 as 0 < a < (3.
References 1. A. Alema and A. G. Siskakis, Integral operators on Hp, Complex Variables, 28 (1995):107-152. 2. R. Yoneda, Multiplication operators, integration operators and companion operators on weighted Bloch spaces, Hokkaido Math. J. 34(2005):135-147. 3. R. Yoneda, Pointwise multipliers from BMOAa to BMOA0, Complex Variables 49 (2003):1045-1061. 4. X. J. Zhang, Extend cesaro operator on the Dirichlet type spaces and Bloch type spaces of C n (in Chinese), Chinese Ann. Math. 26A (2005):139-150. 5. K. H. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York and Basel, 1990.
T H E G R O W T H OF ANALYTIC F U N C T I O N S OF INFINITE O R D E R R E P R E S E N T E D B Y LAPLACE-STIELTJES TRANSFORMATIONS
LINA SHANG LMIB and Department of Mathematics Beihang University Beijing, 100083 P. R. China E-mail: [email protected] ZONGSHENG GAO LMIB and Department of Mathematics Beihang University Beijing, 100083 P. R. China E-mail: [email protected]
In this paper, we investigate the growth and regular growth of functions of infinite order defined by Laplace-Stieltjes transformations and obtain some necessary and sufficient conditions.
Keywords: Laplace-Stieltjes transformation, order, analytic function. 2000 Mathematics Subject Classification: 30D99; 44A10
1. Introduction Many problems about the growth and the value distribution of analytic functions defined by Laplace-Stieltjes transformations have been studied and some important results have been obtained in [1],[2]. But the infinite order of Laplace-Stieltjes transformation hasn't been discussed yet. In this paper, we study the analytic functions of infinite order defined by LaplaceStieltjes transformations in the half right plane and get some results on their growth and regular growth. 231
232
Consider the Laplace-Stieltjes transformation /•OO
/ e~sxda(x) {s = a + it) (1) Jo where a(x) is a complex valued function on x > 0 and a bounded variation function on arbitrary closed interval [0,X] (0 < X < +oo). Choose a sequence {An} such that: 0 = Ai < A2 < • • • < A„ T oo, (2) F(a)=
Inn lim —— = / < oo, n—too
(3)
ln\n
lim (A„ +1 - An) < oo.
(4)
n—*oo
If the transformation (1) satisfies
TET^=0,
(5)
where A* =
sup
| /
An<x
e-^dafo)!, J\n
it follows from [l,Theoreml.3] that the abscissa of uniform convergence of (1) is 0 and the transformation (1) defines an analytic function F(s) in the right-half plane. Set M(a,F)=
sup
\F(a + it)\,
—oo
M U ((7,F)=
\fXe-(°+«)y da{y)\,
sup 0<x<+oo,-oo
li(a, F) = max{A*ne-x"a}, n
Jo
n(a) = max{fc| fi(a, F) =
Ale~Xka}.
k
Definition: T = lim O--.+0
ln+ln+M(<j,F) ——ln+ln+Mu(a,F) : , Tu = lim : , — Ina
CT->+O
— Ina
ln+ln+fi(a, F) TM = lim ; . cr-^+o — Ina We call T order(R) of F(s) in a > 0. Note: It is obvious that M{a,F) < Mu(a,F). Specially, if r = oo, we have TU = oo.
233
2.
Main Results
In this paper we use C for a numerical constant. It will not be the same at each occurrence but it is always independent of all variables. Lemma 1. Suppose that the function
-ln+ln+(p(o-) ;
CT-»+O
= oo.
—Ina
Then there exists a function T(U) such that 1) T(U) is continuous on u > 0 and lim T(U) = oo; u—»+oo 2
= l Wkere
)J™ooT^
t/
>
V-+o
M=
wT(
")>
U
' = U+l^UM
°rU-lniku)'
InU(^)
Proof When u' = u + ln^,u\ , the lemma is given in [3,Theorem4.3]. Imitating the method of [3,Theorem4.3], we can prove the lemma holds when u' = u + j^m^x(c > 0). To finish the proof, we need show that the lemma holds when u' = u— ln{jtu\ • For sufficiently large u and v! = u— t n I // u \, there exists c > 0, such that u = «'(1 - T - ^ ) -
lnU(u)
1
< «'(1 + -j-^r)
lnU(u)
< u'(l + T - y ^ r ) .
lnU{u')
Consequently,
Noticing that x <
lnU(u) < lnU(u'(l + j^fa)) lnU(u') ~ lnU{u')
we derive ,. ^U{u) hm , TT; = 1. «->+(» InUyw) Hence the lemma. Lemma 2. Suppose that {A„} satisfies (2),(3),(4) and the transformation (1) satisfies (5). Then lim
-ln+ln+Mu(a,F) _ ——ln+ln+n(o-, F) , r "\ ' = 1 «=> lim , rr 7 v ' ' = 1,
234
where U is a function satisfying conditions 1), 2) in Lemma 1. Proof Let I{x;a + it) = f Jo
e-{
then for x > An, we have £
e-^daiy)
= / £ e " % J ( y ; a + it) = I(y; a + it)e°y\%n - a / £ e^I(y-
a + it)dy.
By (4), there exists K > 0 such that for n G N, Xn+i — An < K. Furthermore, for sufficiently small cr, we have eKa < 2. Hence, when An < X < A n +i, | /
e-ityda(y)\
< 2Mu(a,F)eX(T
< 2Mu(a,F)ex»+ia
AMu{a,F)ex^.
<
Consequently H(a,F)<4Mu(a,F).
(6)
On the other hand, notice that
f0x e-^^My)
= E 1 AA;+1
e-^yda(y)
fe=i
[I) +it
+ H e-^ ^da(y)
(A„ < x < A n + 1 ).
Let [ e-ityda{y)
Ik(x;it)=
(Afc < x <
Xk+1).
(8)
J\k
Then for Afc < x < Xk+i, - c o < t < +oo, \Ik(x;it)\
(9)
By (7) and (8), for VO < e < 1, A„ < x < A n + i, a > 0, /0X e - ^ + ^ d a f e ) = " E J^+1
e-°ydyIk(y;
it) + / £ e - " % / n ( y ; it)
fe=i
= " E l e - ^ + ^ / ^ A f c + i ; rt) +
1
e-^7fc(y; it)dy]
+ e- I
235
Combine this with (9) and put A = jx e-(»+it)yda(y^
< £
1
/i (A,F)e
"
\J[\<,G
to obtain
A A
* (e- Afc + lCT + e~Xka - e~A*+1<7)
fc=i
+/x(A, F)ex"A(e-xrr = £
/ i(A,F)e
+ e'^"
e~xa)
-
1+, t
" '<±> < fi(A,F)
e1+,nJ(-).
£
fe=i fc=i
(10) Since (3), for any £ > 0, there exists N > 0 such that for n > N, Xn > n<+? > Zn3n. Let T = exp{ ^'}i. Combine with (10) to obtain [T] 2 £ n-in n("/[l+'nl>(±)]) + n=JV+l
Mu{
<MA,F)(C + 2exp((H±«il)l
X
oo £ n~2) n=[T]+l
2)).
Consequently /n + /n + M u ((T,F) < Zn + /n+/x(A,F) + £inZnl7(-) + \ln- + C. 2 cr 2 cr By the property of U(^), we have ——ln+ln+fi(a,F)
ln+ln+Mu(a,F)
Combining this with (6), we conclude Lemma 2. Theorem 1. Suppose that {A„} satisfies (2), (3), (4) and the transformation (1) satisfies (5) and -ln+ln+M(a,F) lim ———- = oo. cr->+o —Incr Then lim
-ln+ln+Mu(a,F) n , TT,i\ '
= 1 < = • lim t n = 1,
lnU{ — )
n—>oo
where =
f InXn/lnUijfe)
" 1
(A*n>l)
0"
«<1)
and U is a function satisfying conditions 1), 2) in Lemma 1.
(11)
236
Proof First we prove that if lim tn = 1, then n—»oo
hm
,
rr/1;
< 1.
(12)
For any 77 > 0, there exists a positive integer N > 0 such that for n > N and A*n > 1,
Let v = U(u) and u = (p(v) be two reciprocally inverse functions. Then for n> N and A*n > 1, we have
A
^
- \na].
(13)
Evidently (13) holds when A*n < 1. Hence we have (13) for Vn > AT. Fix a > 0 and take A such that
Then A = [tf (£)] 1 + , ? + o ( 1 ) (a -> 0+). By (13), when An > A and n > N, A*ne~^
< e x p [ A „ ( - ^ ^ - a)} = exp[
~A"^
Then for sufficiently small
(a -> +0).
Notice that 77 can be arbitrarily small to obtain llm
,
Tr/1.
Applying Lemma 2, hence (12) holds.
< 1-
] < 1.
237
Now we prove that if lim tn = 1, then we cannot have n—+00
lim
-ln+ln+Mu(a,F) " \ ' ' = c < 1.
(14)
Suppose that (14) would hold, choose e > 0 and c + 2e < 1. Then there would exist <7o(0 < a < 1) such that for 0 < a < <7o, 1,
I n M > , f ) < [ l / ( - ) fic+e (7
and consequently for n = 0,1,2, JnA„ - A„a < [U(-)]c+e.
(15)
<7
On the other hand, there exist arbitrarily large integers n such that An > Pij^)}1-6
(K > 1).
(16)
Take such sufficiently large n and take
A. / " , A- ,•
(17)
Combining (15) and (17), we see that for any rj > 0, there are arbitrarily large n and a e (0,00) such that lnA*nl
a
lnU{j^y
and consequently
^1' < P<^"lrt-
(18)
By (17) and (18) we have, for those sufficiently large n for which (16) holds,
An < [ t / ( 7 ^ ) ] ( 1 + " ) ( c + £ ) 7 ^ n t / ( - V ) . And since (16), for the other sufficiently large n, tn < 1 — e. Therefore, lim tn < 1, which is contrary to the hypothesis. n—>oo
The sufficiency of the condition (11) is proved. By the prove of sufficiency we can prove necessity easily. Theorem 2. Suppose that {Xn} satisfies (2), (3), (4) and the transformation (1) satisfies (5) and lim
-ln+ln+M{o,F)
CT^+O
— Ina
= 00.
238
Then ,.
ln+ln+Mu(a,F)
cr—>+0
...-
InU {•L)
,
N
n—»oo
(ii) i/iere exists a non-decreasing positive integer sequence {nv} lim tnv = 1,
lim ^ ± i
n—>oo
v—>oo
satisfying
= 1.
(20)
ln\nv
where =
(lnXn/lnU(j^)
«>1)
0
(A;<1)
" 1
and U is a function satisfying conditions 1), 2) in Lemma 1. Proof First, we prove that the conditions are sufficient. From (ii), for any e > 0 and sufficiently large v, we have lnA*nve-x^°
> — % - -
Xnva,
v(A„r) where <^ is the verse function of U. Take o~v such that -(1-r7^TT)
=
^ ^ -
then {crv} is a decreasing sequence and {o~v} —> 0. If 0 < av < a < crv-i, we have ln+fi(a,F)
Noticing that lim V—*0O
llim im — -—
> ln+jj,(av-i,
F) >
"T1 - A ¥>(*£3i)
"" +1 = 1 and InXr = lim = 11U1
;
;—i ;
i
—
= lim —f~T > 1™ , TT/ T , > 1, v^oolnUt^—) ~ v^oolnUi-^) ~ we have lim
-if^- = 1.
239
Therefore, by the property of U(~) we obtain ln+ln+u,(a,F) .. ln+ln+u(a,F) ^ ,. ln+ln+u{av-i,F) TTT—- = hm — —\—-— > hm —;— T^o lnU(±) v^°° InU(^) ~ «^°o j n £/(_L_) hm
^ , >
1-e.
Using (6), .. hm o—>o
ln+ln+Mu(a,F) > 1 - £. lnU{±)
Combining this with Theorem 1, we obtain the sufficiency. Now we prove the necessity of the conditions. The conclusion (i) has been proved in Theorem 1. Here we prove that (ii) is true. Choose {e p }(0 < £ p < 1) such that e p | 0(p —* oo). Put Ep = {n:tn
>l-ep}.
(21)
From the right hand side of (19), we can see that Ep is an unempty infinite set and Ep+\ c Ep for each p. Arrange the elements of Ep in an increasing order and denote it by {n„ }. ln\
(p )
1) If lim , "" +1 = 1 for all p G N, there exists N„ G Ev(p = 1,2, • • •), such that for nvp' > Np,
InX ( p ) T-r^
+ eP.
(22)
Since 2?p+i C J?p, we can suppose Np+\ > Np. Put E'p = {n £ Ep : Np < n < Np+i}. Then the elements of E'p can satisfy (21) and (22). Finally arrange the elements of E = (I ££ in an increasing order and denote it by {nv}. It is easy to see that {nv} satisfy (20). The necessity is proved. 2) If there exists certain p G N satisfying
lim . . " +1 ^ 1, then v—>oo
InX
(p)
1
mA
p)
_(p)
lim , .""+ > 1. Thus there must exist a subset of | n i } (for convenience we still denote it by {nvp }), satisfying /nA (p)
y/nA^ p)> l (
+
7
(v = l , 2 , . . . ) ,
constant. Put (p)
Tli
(P)
'
(P)
' " ) nu — "
(P)
- n2 , n2 — n4 ,
"
(p) n 1v-\i (p)
240
Then {nv}, {nv} are increasing positive integer sequences which satisfy "v < n'v+i, K>; > ^
(« = 1,2,---).
(23)
Put (}=?£. For sufficiently large v, when nv < n < nv, we have tn < 1 — e p < 1 — j3. Consequently lnA*ne-^°
<
X
\_
- \na,
(24)
where
