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George A. Anastassiou
Inequalities Based on Sobolev Representations
123
George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152 USA
[email protected]
ISSN 2191-8198 e-ISSN 2191-8201 ISBN 978-1-4614-0200-8 e-ISBN 978-1-4614-0201-5 DOI 10.1007/978-1-4614-0201-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011929870 Mathematics Subject Classification (2010): 26D10, 26D15, 26D20 c George A. Anastassiou All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To the memory of my friend Khalid Khouri who left this world too young The measure of success for a person is the magnitude of his/her ability to convert negative conditions to positive ones and achieve goals —The author
Preface
This brief monograph is the first one to deal exclusively with very general tight integral inequalities of Chebyshev–Gr¨uss, Ostrowski types, and of the comparison of integral means. These rely on the well-known Sobolev integral representations of functions. The inequalities engage ordinary and weak partial derivatives of the involved functions. Applications of these developments are illustrated. On the way to prove the main results we derive important estimates for the averaged Taylor polynomials and remainders of Sobolev integral representations. The exposed results expand to all possible directions. We examine both the univariate and multivariate cases. For the convenience of the reader, each chapter of this book is written in a selfcontained style. This treatise relies on the author’s last year of related research work. Advanced courses and seminars can be taught out of this brief book. All necessary background and motivations are given in each chapter. A related list of references is also given at the end of each chapter. These results first appeared in my articles that are mentioned in the references. The results are expected to find applications in many subareas of mathematical analysis, inequalities, partial differential equations, information theory, etc. As such this monograph is suitable for researchers, graduate students, seminars of the above subjects, and also for all science libraries. The preparation of this booklet took place during 2010–2011 in Memphis, TN, USA. I thank my family for their dedication and love to me, which was the strongest support during the writing of this book. Department of Mathematical Sciences The University of Memphis, TN, USA March 5, 2011
George A. Anastassiou
vii
Contents
1
Univariate Integral Inequalities Based on Sobolev Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 1 1.2 Background.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 2 1.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 11 1.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 32 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 34
2 Multivariate Integral Inequalities Deriving from Sobolev Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 2.2 Background.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 2.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .
35 35 36 50 64 65
ix
Chapter 1
Univariate Integral Inequalities Based on Sobolev Representations
Here we present very general univariate tight integral inequalities of Cheby-shev– Gr¨uss, Ostrowski types for comparison of integral means and information theory. These are based on the well-known Sobolev integral representations of a function. The inequalities engage ordinary and weak derivatives of the involved functions. Applications are given. On the way to prove the main results we derive important estimates for the averaged Taylor polynomials and remainders of Sobolev integral representations. The results are explained thoroughly. This chapter relies on [4].
1.1 Introduction This chapter is greatly motivated by the following theorems: Theorem 1.1 (Chebychev, 1882, [7]). Let f; gW Œa; b ! R absolutely continuous functions. If f 0 ; g0 2 L1 .Œa; b/, then ˇ ! Z !ˇ Z b ˇ 1 Z b ˇ b 1 ˇ ˇ f .x/ g .x/ dx f .x/ dx g .x/ dx ˇ ˇ ˇb a a ˇ .b a/2 a a
1 .b a/2 f 0 1 g 0 1 : 12
(1.1)
Theorem 1.2 (G. Gruss, ¨ 1935, [11]). Let f; g integrable functions from Œa; b ! R, such that m f .x/ M , g .x/ , for all x 2 Œa; b, where m; M; ; 2 R. Then ˇ ! Z !ˇ Z b ˇ 1 Z b ˇ b 1 ˇ ˇ f .x/ g .x/ dx f .x/ dx g .x/ dx ˇ ˇ 2 ˇb a a ˇ .b a/ a a 1 .M m/ . / : 4 In 1938, A. Ostrowski [14] proved.
G.A. Anastassiou, Inequalities Based on Sobolev Representations, SpringerBriefs in Mathematics 2, DOI 10.1007/978-1-4614-0201-5 1, © George A. Anastassiou
(1.2)
1
2
1 Univariate Integral Inequalities Based on Sobolev Representations
Theorem 1.3. Let f W Œa; b ! R be continuous on Œa; b and differentiable on .a; b/ whose derivative f 0 W .a; b/ ! R is bounded on .a; b/, i.e., kf 0 k1 D sup jf 0 .t/j < C1: Then t 2.a;b/
ˇ ˇ " 2 # ˇ 1 Z b ˇ x aCb 1 ˇ ˇ 2 C .b a/ f 0 1 ; f .t/ dt f .x/ˇ ˇ 2 ˇb a a ˇ 4 .b a/ for any x 2 Œa; b. The constant
1 4
(1.3)
is the best possible.
See also [1–3] for related works that inspired as well this chapter. In this chapter using the univariate Sobolev-type representation formulae, see Theorems 1.10, 1.14 and also Corollaries 1.11, 1.12, we first estimate their remainders and then the involved averaged Taylor polynomials. Based on these estimates we derive lots of very tight inequalities on R: of Chebyshev–Gr¨uss type, Ostrowski type, for comparison of integral means and Csiszar’s f -divergence with applications. The results involve ordinary and weak derivatives and they go to all possible directions using various norms. All of our tools come from the excellent monograph by V. Burenkov, [6].
1.2 Background Here we follow [6]. For a measurable nonempty set ˝ Rn , n 2 N we shall denote by Lloc p .˝/ (1 p 1) – the set of functions defined on ˝ such that for each compact K ˝; f 2 Lp .K/. Definition 1.4. Let ˝ Rn be an open set, ˛ 2 ZnC , ˛ ¤ 0 and f; g 2 Lloc 1 .˝/. The function g is a weak derivative of the function f of order ˛ on ˝ (briefly g D Dw˛ f ) if 8 ' 2 C01 .˝/ (i.e., ' 2 C 1 .˝/ compactly supported in ˝) Z Z j˛j ˛ fD ' dx D .1/ g' dx: (1.4) ˝
˝
Definition 1.5. Wpl .˝/ (l 2 N, 1 p 1) – Sobolev space, which is the Banach space of functions f 2 Lp .˝/ such that 8 ˛ 2 ZnC where j˛j l the weak derivatives Dw˛ f exist on ˝ and Dw˛ f 2 Lp .˝/, with the norm kf kWpl .˝/ D
X D ˛ f w
j˛jl
Lp .˝/
:
(1.5)
.loc/ Definition 1.6. For l 2 N, we define the Sobolev-type local space W1l .˝/ WD ff W ˝ ! R W f 2 L1loc .˝/ and all f -distributional partials of orders l belong to L1loc .˝/g D ff 2 Lloc 1 .˝/ W for each open set G compactly embedded into ˝, f 2 W1l .G/g:
1.2
Background
3
We use Definitions 1.4–1.6 on R. Next we mention Sobolev’s integral representation from [6]. Definition 1.7 ([6], p. 82). Let 1 < a < b < 1; Z ! 2 L1 .a; b/ , Define
b
! .x/ dx D 1:
(1.6)
a
( Ry a y x b; a R! .u/ du; .x; y/ WD b y ! .u/ du; a x < y b:
(1.7)
Proposition 1.8 ([6], p. 82). Let f be absolutely continuous on Œa; b. Then 8 x2 .a; b/ Z b Z b f .x/ D f .y/ ! .y/ dy C .x; y/ f 0 .y/ dy; (1.8) a
a
the simplest case of Sobolev’s integral representation. Remark 1.9 ([6], pp. 82–83). We have that is bounded: 8 x; y 2 Œa; b , j .x; y/j k!kL1 .a;b/ ;
(1.9)
and if ! 0, then 8 x; y 2 Œa; b , j .x; y/j .b; b/ D 1: If ! is symmetric with respect to
aCb , 2
then 8 y 2 Œa; b we have
ˇ ˇ ˇ ˇ ˇ a C b ; y ˇ 1 : ˇ ˇ 2 2 Examples of !: ! .x/ D also ! .x/ D
1 ; ba
8x 2 .a; b/ ;
1 .a;aC 1 / C .b 1 ;b/ ; m m 2m
where .˛;ˇ/ denotes the characteristic function of .˛; ˇ/, m 2 N and m 2.b a/1 : loc .a; b/, then f is equivalent to a function, which is locally absoIf f 2 W1l lutely continuous on .a; b/ (its ordinary derivative, which exists almost everywhere on .a; b/, is a weak derivative fw0 of f ). Thus (1.8) holds for almost every x 2 .a; b/ if f 0 is replaced by fw0 :
4
1 Univariate Integral Inequalities Based on Sobolev Representations
P0
kD1
In this chapter sums of the form We mention
D 0:
Theorem 1.10 ([6], p. 83). Let l 2 N, 1 a < ˛ < ˇ < b 1 and
! 2 L1 .R/ , (support) sup p! Œ˛; ˇ ; R R ! .x/ dx D 1:
(1.10)
Moreover, assume that the derivative f .l1/ exists and is locally absolutely continuous on .a; b/. Then 8 x 2 .a; b/ f .x/ D
Z l1 X 1 b .k/ f .y/ .x y/k ! .y/ dy kŠ a kD0
C
1 .l 1/Š
Z
b
.x y/l1 .x; y/ f .l/ .y/ dy;
(1.11)
a
and Z l1 X 1 ˇ .k/ f .x/ D f .y/ .x y/k ! .y/ dy kŠ ˛ kD0
1 C .l 1/Š
Z
bx
.x y/l1 .x; y/ f .l/ .y/ dy;
(1.12)
ax
where ax D x, bx D ˇ for x 2 .a; ˛; ax D ˛, bx D ˇ for x 2 .˛; ˇ/; ax D ˛, bx D x for x 2 Œˇ; b/. If, in particular, 1 < a < b < 1, f .l1/ exists and is absolutely continuous on Œa; b, then (1.11), (1.12) hold 8 x 2 Œa; b and for any interval .˛; ˇ/ .a; b/ : Corollary 1.11 ([6], p. 85). Assume that l > 1, condition (1.10) is replaced by
! 2 C .l2/ .R/ , sup p! Œ˛; ˇ ; R R ! .x/ dx D 1;
(1.13)
and the derivative ! .l2/ is absolutely continuous on Œa; b : Then for the same f as in Theorem 1.10, 8 x 2 .a; b/ Z f .x/ D ˛
ˇ
l1 X .1/k h kD0
1 C .l 1/Š
kŠ Z
bx
ax
! i.k/ .x y/ ! .y/ f .y/ dy k
y
.x y/l1 .x; y/ f .l/ .y/ dy:
(1.14)
1.2
Background
5
In particular, here ! .˛/ D ::: D ! .l2/ .˛/ D ! .ˇ/ D ::: D ! .l2/ .ˇ/ D 0:
(1.15)
Corollary 1.12 ([6], p. 86). Assume that l; m 2 N, m < l. Then for the same f and ! as in Corollary 1.11, 8 x 2 .a; b/ f
.m/
Z .x/ D
ˇ
lm1 X
˛
kD0
! i.kCm/ .1/kCm h k .x y/ ! .y/ f .y/ dy y kŠ
1 C .l m 1/Š
Z
bx
.x y/lm1 .x; y/ f .l/ .y/ dy: (1.16)
ax
Remark 1.13 ([6], p. 86). The first summand in (1.14) can take the form: 8 R P ˇ l1 s .s/ ˆ .x y/ ! .y/ f .y/ dy; s < ˛ sD0 s P ls1 s C k .1/ ˆ : : where s WD sŠ kDs k
(1.17)
Similarly, we have for the first summand of (1.16) the following form: 8 R P ˇ l1 sm .s/ ˆ ! .y/ f .y/ dy; < ˛ sDm s;m .x y/ sCk .1/s Pls1 ˆ : : where s;m WD .sm/Š kDs k
(1.18)
We need Theorem 1.14 ([6], p. 91). Let l 2 N , 1 a < ˛ < ˇ < b 1, ! satisfy condition Z ! .x/ dx D 1 (1.19) ! 2 L1 .R/ ; sup p! Œ˛; ˇ ; R
loc .a; b/ : Then for almost every x 2 .a; b/ and f 2 W1l f .x/ D
Z l1 X 1 ˇ .k/ f .y/ .x y/k ! .y/ dy kŠ ˛ w kD0
1 C .l 1/Š where ax , bx as in Theorem 1.10. We denote fw.0/ WD f:
Z
bx ax
.x y/l1 .x; y/ fw.l/ .y/ dy;
(1.20)
6
1 Univariate Integral Inequalities Based on Sobolev Representations
Remark 1.15 ([6], p. 92). By Theorem 1.14 it follows that if in Corollaries 1.11, loc .a; b/ then equalities (1.14) and (1.16) hold almost everywhere 1.12 f 2 W1l on .a; b/, if we substitute f .l/ , f .m/ by the weak derivatives fw.l/ ; fw.m/ ; respectively. Next we estimate the remainders of the above mentioned Sobolev representations. We make Remark 1.16. Denote by f m 2 ZC . We estimate
.k/
either f .k/ or fw.k/ , where k 2 N. Let 0 m < l,
1 Rm;l f .x/ WD .l m 1/Š
Z
ˇ
.x y/lm1 .x; y/ f
.l/
.y/ dy;
(1.21)
˛
for x 2 .˛; ˇ/, where as in (1.7), see also (1.9). So we obtain R0;l f .x/ WD
1 .l 1/Š
Z
ˇ
.x y/l1 .x; y/ f
.l/
.y/ dy:
(1.22)
˛
Thus we derive Z ˇ k!kL1 .a;b/ .ˇ ˛/lm1 ˇ ˇˇ .l/ ˇ ˇf .y/ˇ dy .l m 1/Š ˛ .l/ .ˇ ˛/lm1 k!kL1 .a;b/ f L1 .˛;ˇ/ ; D .l m 1/Š
jRm;l f .x/j
x 2 .˛; ˇ/ : We also have jRm;l f .x/j If f
.l/
Z
k!kL1 .a;b/ .l m 1/Š
˛
ˇ
ˇ .l/ ˇ ˇ ˇ jx yjlm1 ˇf .y/ˇ dy DW I1 :
2 L1 .˛; ˇ/, then
I1
.l/ k!kL1 .a;b/ f
L1 .˛;ˇ/
.l m 1/Š
Z
ˇ
! jx yj
lm1
dy :
˛
But Z
ˇ
jx yj
lm1
Z
x
dy D
˛
˛
D
.x y/
lm1
Z dy C
ˇ
x
.ˇ x/lm C .x ˛/lm : l m
.y x/lm1 dy
(1.23)
1.2
Background
Therefore if f
.l/
7
2 L1 .˛; ˇ/, then
jRm;l f .x/j
.l/ k!kL1 .a;b/ f
L1 .˛;ˇ/
.l m/Š
x 2 .˛; ˇ/ : Let now p; q > 1 W
1 p
1 q
C
D 1: If f Z
k!kL1 .a;b/ I1 .l m 1/Š
ˇ
.l/
.ˇ x/lm C .x ˛/lm ;
(1.24)
2 Lp .˛; ˇ/, then ! q1
jx yj
q.lm1/
.l/ f
dy
Lp .˛;ˇ/
˛
:
However Z
ˇ
q.lm1/
jx yj
Z
x
dy D
˛
.x y/
q.lm1/
Z dy C
˛
D Hence if f
.l/
ˇ
.y x/q.lm1/ dy
x
.x ˛/q.lm1/C1 C .ˇ x/q.lm1/C1 : q .l m 1/ C 1
2 Lp .˛; ˇ/, then
jRm;l f .x/j
.l/ k!kL1 .a;b/ f
Lp .˛;ˇ/
.l m 1/Š .ˇ x/q.lm1/C1 C .x ˛/q.lm1/C1 q .l m 1/ C 1
! q1 ;
(1.25)
x 2 .˛; ˇ/ : If sup p! Œ˛; ˇ, then k!kL1 .a;b/ D k!kL1 .˛;ˇ/ :
(1.26)
If ! 2 C .R/ and sup p! Œ˛; ˇ, then k!kL1 .˛;ˇ/ k!k1;Œ˛;ˇ .ˇ ˛/ :
(1.27)
8
1 Univariate Integral Inequalities Based on Sobolev Representations
We make Remark 1.17. Here we estimate from the Taylor’s averaged polynomial, see (1.12) and (1.20), that part Q
l1
Z l1 X 1 ˇ .k/ f .x/ WD f .y/ .x y/k ! .y/ dy; kŠ ˛
(1.28)
kD1
called also quasi-averaged Taylor polynomial. When l D 1, then Q0 f .x/ D 0: We see that Z l1 ˇ ˇ l1 ˇ X 1 ˇ ˇˇ .k/ ˇ k ˇQ f .x/ˇ ˇf .y/ˇ jx yj j! .y/j dy kŠ ˛ kD1 ! l1 X .ˇ ˛/k .k/ k!kL1 .R/ ; f L1 .˛;ˇ/ kŠ
(1.29)
kD1
given that k!kL1 .R/ < 1, x 2 .˛; ˇ/. Similarly, when f obtain ˇ ˇ l1 ˇQ f .x/ˇ
l1 X
.k/
2 L1 .˛; ˇ/, k D 1; :::; l 1, and k!kL1 .R/ < 1 we
.k/ f
kD1
L1 .˛;ˇ/
kŠ
k!kL1 .R/ Z
ˇ
l1 X .ˇ x/kC1 C .x ˛/kC1 D .k C 1/Š kD1
jx yjk dy
˛
!
.k/ f
L1 .˛;ˇ/
! k!kL1 .R/ ; (1.30)
x 2 .˛; ˇ/. .k/ 2 Lp .˛; ˇ/, k D 1; :::; l 1; and again Let p; q > 1 W p1 C q1 D 1; f k!kL1 .R/ < 1. Then .k/ ! q1 Z ˇ l1 ˇ l1 ˇ X f Lp .˛;ˇ/ kq ˇQ f .x/ˇ jx yj dy k!kL1 .R/ kŠ ˛ kD1
.k/ 0 1 ! q1 f l1 .kqC1/ .kqC1/ C .x ˛/ Lp .˛;ˇ/ C BX .ˇ x/ D@ A k!kL1 .R/ ; kq C 1 kŠ kD1
(1.31) x 2 .˛; ˇ/ :
1.2
Background
9 .k/
Suppose ! 2 L1 .R/ and f
2 L1 .˛; ˇ/, k D 1; :::; l 1; then
.k/ 1 k l1 f .ˇ ˛/ X ˇ l1 ˇ L1 .a;ˇ/ C ˇQ f .x/ˇ B @ A k!kL1 .R/ ; kŠ 0
(1.32)
kD1
x 2 .˛; ˇ/. Suppose p; q > 1 W then
1 C q1 p
.k/
D 1; f
2 Lp .˛; ˇ/, k D 1; :::; l 1I ! 2 Lq .˛; ˇ/,
! l1 X ˇ l1 ˇ .ˇ ˛/k .k/ ˇQ f .x/ˇ k!kLq .˛;ˇ/ ; f Lp .˛;ˇ/ kŠ
(1.33)
kD1
x 2 .˛; ˇ/. Suppose p; q; r > 1 W ! 2 Lq .˛; ˇ/, then 0 ˇ l1 ˇ ˇQ f .x/ˇ B @
l1 X
1 p
C
1 q
C
1 r
.k/ f
D 1; f
Lp .˛;ˇ/
kŠ
kD1
.k/
2 Lp .˛; ˇ/, k D 1; :::; l 1I
.ˇ x/.krC1/ C .x ˛/.krC1/ .kr C 1/
k!kLq .˛;ˇ/ ;
! 1r
1 C A (1.34)
x 2 .˛; ˇ/: We also make
R Remark 1.18. Here l > 1, ! 2 C .l2/ .R/, sup p! Œ˛; ˇ, R ! .x/ dx D 1, and the derivative ! .l2/ is absolutely continuous on Œa; b. Hence we have that Ql1 f .x/ D
Z l1 X .1/k kD1
kŠ
ˇ
˛
h
i.k/ f .y/ dy; .x y/k ! .y/ y
(1.35)
8 x 2 .˛; ˇ/. And it holds l1 ˇ l1 ˇ X 1 ˇQ f .x/ˇ kŠ kD1
8 x 2 .˛; ˇ/.
Z ˛
ˇ
ˇh i.k/ ˇˇ ˇ ˇ .x y/k ! .y/ ˇ jf .y/j dy; ˇ y ˇ
(1.36)
10
1 Univariate Integral Inequalities Based on Sobolev Representations
Consequently, 8 x 2 .˛; ˇ/, ˇ l1 ˇ ˇQ f .x/ˇ 8 h i.k/ Pl1 1 ˆ k ˆ kf kL1 .˛;ˇ/ ; ˆ kD1 kŠ .x y/ ! .y/ ˆ ˆ y 1 ˆ ˆ ˆ ˆ if f 2 L1 .˛; ˇ/; ˆ ! ˆ h ˆ i.k/ ˆ P ˆ l1 k 1 ˆ ˆ kf kL1 .˛;ˇ/ ; ˆ kD1 kŠ .x y/ ! .y/ < y L .˛;ˇ/ 1 if f 2 L1 .˛; ˇ/; ˆ ˆ ˆ ˆ when p; q > 1 W p1 C q1 D 1, we have ˆ ˆ ! ˆ h ˆ i.k/ ˆ Pl1 1 ˆ k ˆ ˆ kf kLp .˛;ˇ/ ; ˆ kD1 kŠ .x y/ ! .y/ ˆ y L .˛;ˇ/ ˆ ˆ q ˆ : if f 2 Lp .˛; ˇ/: Let l; m 2 N, m < l, and f; ! as above, x 2 .˛; ˇ/. We consider here lm1 i.kCm/ X .1/kCm Z ˇ h k l1 Qm f .x/ WD f .y/ dy: .x y/ ! .y/ y kŠ ˛
(1.37)
(1.38)
kD1
l1 f .x/ WD 0: When l D m C 1, then Qm Hence it holds i.kCm/ ˇˇ X 1 Z ˇ ˇˇh ˇ lm1 ˇ l1 ˇQ f .x/ˇ ˇ .x y/k ! .y/ ˇ jf .y/j dy; m ˇ y kŠ ˛ ˇ
(1.39)
kD1
8 x 2 .˛; ˇ/ : Consequently, 8 x 2 .˛; ˇ/, ˇ l1 ˇ ˇQ f .x/ˇ m h 8 i.kCm/ Plm1 1 ˆ k ˆ .x y/ ! .y/ kf kL1 .˛;ˇ/ ; ˆ kD1 ˆ kŠ ˆ y ˆ 1 ˆ ˆ ˆ if f 2 L1 .˛; ˇ/; ˆ ! ˆ h ˆ i.kCm/ ˆ P ˆ lm1 k 1 ˆ ˆ ! .y/ kf kL1 .˛;ˇ/ ; ˆ kD1 kŠ .x y/ < y L1 .˛;ˇ/ if f 2 L1 .˛; ˇ/; ˆ ˆ ˆ ˆ when p; q > 1 W p1 C q1 D 1, we have ˆ ˆ ! ˆ h ˆ i.kCm/ ˆ Plm1 1 ˆ k ˆ ˆ .x y/ ! .y/ kf kLp .˛;ˇ/ ; ˆ kD1 kŠ ˆ y ˆ Lq .˛;ˇ/ ˆ ˆ : if f 2 Lp .˛; ˇ/:
(1.40)
1.3
Main Results
11
We also need .k/
Remark 1.19. Here again f means either f .k/ or fw.k/ , k 2 N. We rewrite (1.12), (1.14), and (1.20). For x 2 .˛; ˇ/ we get Z
ˇ
f .x/ D ˛
f .y/ ! .y/ dy C Ql1 f .x/ C R0;l f .x/ :
(1.41)
Also for x 2 .˛; ˇ/ we rewrite (1.16) (see also Remark 1.15) as follows: f
.m/
.x/ D .1/m
Z
ˇ
˛
l1 f .y/ ! .m/ .y/ dy C Qm f .x/ C Rm;l f .x/ :
(1.42)
1.3 Main Results On the way to prove the general Chebyshev–Gr¨uss-type inequalities, we establish the general Theorem 1.20. For f; g under the assumptions of any of Theorem 1.10, Corollary 1.11 and Theorem 1.14 we obtain that ˇZ ! Z !ˇ Z ˇ ˇ ˇ ˇ ˇ ˇ ˇ .f; g/ WD ˇ ! .x/ f .x/ g .x/ dx ! .x/ f .x/ dx ! .x/ g .x/ dx ˇ ˇ ˇ ˛ ˛ ˛ " Z ! Z ˇ ˇ ˇ l 1 ˇ ˇ l 1 ˇ 1 ˇ ˇ ˇ ˇ j! .x/j jg .x/j Q f .x/ dx C j! .x/j jf .x/j Q g .x/ dx 2 ˛ ˛ !# Z ˇ Z ˇ ˇ ˇ ˇ ˇ : j! .x/j jg .x/j ˇR0;l f .x/ˇ dx C j! .x/j jf .x/j ˇR0;l g .x/ˇ dx C ˛
˛
(1.43)
Proof. For x 2 .˛; ˇ/ we have Z
ˇ
f .x/ D ˛
and
Z g .x/ D ˛
ˇ
f .y/ ! .y/ dy C Ql1 f .x/ C R0;l f .x/;
g .y/ ! .y/ dy C Q l1 g .x/ C R0;l g .x/:
Hence
Z
ˇ
f .y/ ! .y/ dy
! .x/ f .x/ g .x/ D ! .x/ g .x/ ˛
C ! .x/ g .x/ Ql1 f .x/ C ! .x/ g .x/ R0;l f .x/;
12
1 Univariate Integral Inequalities Based on Sobolev Representations
and Z
ˇ
! .x/ f .x/ g .x/ D ! .x/ f .x/
g .y/ ! .y/ dy
˛
C ! .x/ f .x/ Ql1 g .x/ C ! .x/ f .x/ R0;l g .x/ : Therefore Z
Z
ˇ
! Z
ˇ
! .x/ f .x/ g .x/ dx D
! .x/ g .x/ dx
˛
!
ˇ
f .x/ ! .x/ dx
˛
˛
Z
ˇ
C
! .x/ g .x/ Ql1 f .x/ dx
˛
Z
ˇ
C ˛
! .x/ g .x/ R0;l f .x/ dx;
and Z
ˇ
Z
ˇ
! .x/ f .x/ g .x/ dx D
˛
! Z ! .x/ f .x/ dx
˛
! g .x/ ! .x/ dx
˛
Z
ˇ
C Z
ˇ
! .x/ f .x/ Ql1 g .x/ dx
˛ ˇ
C ˛
! .x/ f .x/ R0;l g .x/ dx:
Consequently it holds Z
ˇ
Z ! .x/ f .x/ g .x/ dx
˛
ˇ
! Z ! .x/ f .x/ dx
˛
Z
ˇ
D
ˇ
! g .x/ ! .x/ dx
˛
! .x/ g .x/ Q l1 f .x/ dx C
˛
Z
ˇ
! .x/ g .x/ R0;l f .x/ dx;
a
and Z
ˇ
Z ! .x/ f .x/ g .x/ dx
˛
ˇ
! Z ! .x/ f .x/ dx
˛
Z D ˛
ˇ
ˇ
! .x/ f .x/ Q l1 g .x/ dx C
! g .x/ ! .x/ dx
˛
Z ˛
ˇ
! .x/ f .x/ R0;l g .x/ dx:
1.3
Main Results
13
Adding the last two equalities and dividing by two, we get Z
ˇ
Z
ˇ
! .x/ f .x/ g .x/ dx
˛
1 D 2
" Z
! Z ! .x/ f .x/ dx
˛ ˇ
! .x/ g .x/ Q
l1
ˇ
! g .x/ ! .x/ dx
˛
Z f .x/ dx C
˛
!
ˇ
! .x/ f .x/ Q
l1
g .x/ dx
˛
Z C ˛
ˇ
Z ! .x/ g .x/ R0;l f .x/ dx C
ˇ ˛
!# ! .x/ f .x/ R0;l g .x/ dx
;
hence proving the claim. General Chebyshev–Gr¨uss inequalities follow. We give Theorem 1.21. Let f; g with f .l1/ , g.l1/ absolutely continuous R on Œa; b R; l 2 N; .˛; ˇ/ .a; b/ : Let also ! 2 L1 .R/, sup p! Œ˛; ˇ, R ! .x/ dx D 1: Then ˇZ ! Z !ˇ Z ˇ ˇ ˇ ˇ ˇ ˇ ˇ ! .x/ f .x/ g .x/ dx ! .x/ f .x/ dx ! .x/ g .x/ dx ˇ ˇ ˇ ˛ ˇ ˛ ˛ .k/ "" ! l1 f X .ˇ ˛/k k!k2L1 .R/ 1;.˛;ˇ/ kgk1;.˛;ˇ/ 2 kŠ kD1 !# l1 g .k/ X .ˇ ˛/k 1;.˛;ˇ/ C kf k1;.˛;ˇ/ kŠ kD1
" C
kgk1;.˛;ˇ/ f .l/
L1 .a;ˇ/
C kf k1;.˛;ˇ/ g.l/
L1 .˛;ˇ/
.ˇ ˛/l1 .l 1/Š
## :
(1.44)
Proof. By (1.23) and (1.32). Theorem 1.22. Let f; g 2 C l .Œa; R b/, Œa; b R, l 2 N, .˛; ˇ/ .a; b/. Let also ! 2 L1 .R/, sup p! Œ˛; ˇ, R ! .x/ dx D 1: Then ˇZ ! Z !ˇ Z ˇ ˇ ˇ ˇ ˇ ˇ ˇ ! .x/ f .x/ g .x/ dx ! .x/ f .x/ dx ! .x/ g .x/ dx ˇ ˇ ˇ ˇ ˛ ˛ ˛ ( !! " l1 f .k/ X .ˇ ˛/k k!kL1 .R/ 1;.˛;ˇ/ k!kL1 .R/ kgk1;.˛;ˇ/ 2 kŠ kD1
14
1 Univariate Integral Inequalities Based on Sobolev Representations
.k/ !!) l1 g X .ˇ ˛/k 1;.˛;ˇ/
C kf k1;.˛;ˇ/
kŠ
kD1
" C k!k1;.˛;ˇ/
.ˇ ˛/lC1 .l C 1/Š
kgk1;.˛;ˇ/ f .l/
1;.˛;ˇ/
##
C kf k1;.˛;ˇ/ g.l/
:
1;.˛;ˇ/
(1.45)
Proof. By (1.24) and (1.32). We further present loc .a; b/ I a; b 2 RI .˛; ˇ/ .a; b/ ; l 2 NI Theorem 1.23. Let f; g 2 WR1l ! 2 L1 .R/, sup p! Œ˛; ˇ, R ! .x/ dx D 1. Then ˇZ ! Z !ˇ Z ˇ ˇ ˇ ˇ ˇ ˇ ˇ ! .x/ f .x/ g .x/ dx ! .x/ f .x/ dx ! .x/ g .x/ dx ˇ ˇ ˇ ˇ ˛ ˛ ˛
k!k2L1 .R/
(" kgkL1 .˛;ˇ/
2
l1 X .ˇ ˛/k .k/ fw L1 .˛;ˇ/ kŠ
!!
kD1
C kf kL1 .˛;ˇ/
l1 X .ˇ ˛/k .k/ gw L1 .˛;ˇ/ kŠ
!!#
kD1
" C
kgkL1 .˛;ˇ/ fw.l/
L1 .˛;ˇ/
C kf kL1 .˛;ˇ/ gw.l/
L1 .˛;ˇ/
.ˇ ˛/l .l 1/Š
#) :
(1.46)
Proof. By (1.23) and (1.29). loc Theorem 1.24. Let f; g 2 W1l .a; b/ I a; b 2 RI .˛; ˇ/ .a; b/ ; l 2 NI R ! 2 L1 .R/, sup p! Œ˛; ˇ, R ! .x/ dx D 1: Furthermore assume fw.k/ ; gw.k/ 2 L1 .˛; ˇ/ ; k D 1; :::; l: Then ˇZ ! Z !ˇ Z ˇ ˇ ˇ ˇ ˇ ˇ ˇ ! .x/ f .x/ g .x/ dx ! .x/ f .x/ dx ! .x/ g .x/ dx ˇ ˇ ˇ ˇ ˛ ˛ ˛
k!k2L1 .R/ 2
(" kgkL1 .˛;ˇ/
! l1 f .k/ X .ˇ ˛/kC1 w L1 .˛;ˇ/ kD1
kŠ
1.3
Main Results
15
C kf kL1 .˛;ˇ/ " C
!# l1 g .k/ X .ˇ ˛/kC1 w L1 .˛;ˇ/ kŠ
kD1
kgkL1 .˛;ˇ/ fw.l/
L1 .˛;ˇ/
C kf kL1 .˛;ˇ/ gw.l/
L1 .˛;ˇ/
.ˇ ˛/lC1 .l 1/Š
#) : (1.47)
Proof. As in (1.24) and by (1.30). loc .a; b/ I a; b 2 RI .˛; ˇ/ .a; b/ ; l 2 NI Theorem 1.25. Let f; g 2 W1Rl ! 2 L1 .R/, sup p! Œ˛; ˇ, R ! .x/ dx D 1: Furthermore, assume for p > 1 that fw.k/ ; gw.k/ 2 Lp .˛; ˇ/ ; k D 1; :::; l: Then ˇZ ! Z ˇ ˇ ˇ ˇ ! .x/ f .x/ g .x/ dx ! .x/ f .x/ dx .f; g/ WD ˇ ˇ ˛ ˛ !ˇ Z ˇ ˇ ˇ ! .x/ g .x/ dx ˇ ˇ ˛ ! "( 1 l1 X k!k2L1 .R/ .ˇ ˛/kC1 p .k/ kgkL1 .˛;ˇ/ fw Lp .˛;ˇ/ 2 kŠ kD1 !) 1 l1 X .ˇ ˛/kC1 p .k/ C kf kL1 .˛;ˇ/ gw Lp .˛;ˇ/ kŠ kD1 ( C kgkL1 .˛;ˇ/ fw.l/ Lp .˛;ˇ/
C kf kL1 .˛;ˇ/ gw.l/
Lp .˛;ˇ/
1
.ˇ ˛/lC1 p .l 1/Š
)# : (1.48)
Proof. Working as in (1.25) and from (1.33). Remark 1.26. When f; g 2 C l .Œa; b/ ; l 2 N; Theorems 1.23–1.25 are again valid. In this case, we substitute fw.k/ ; gw.k/ by f .k/ , g .k/ in all inequalities (1.46)– (1.48); k D 1; :::; l: We continue with
R Theorem 1.27. Let l 2 N f1g; ! 2 C .l2/ .R/ ; sup p! Œ˛; ˇ, R ! .x/ dx D 1; and the derivative ! .l2/ is absolutely continuous on Œa; b RI .˛; ˇ/ loc .l/ .a; b/, or f; g 2 C l .Œa; b/ : Here f denotes .a; b/ : Here suppose f; g 2 W1l either fw.l/ or f .l/ , and .f; g/ as in (1.43).
16
1 Univariate Integral Inequalities Based on Sobolev Representations
We have the following cases: (i) It holds
" k!k1 2 kgkL1 .˛;ˇ/ kf kL1 .˛;ˇ/ .f; g/ 2 ! h l1 i.k/ X 1 k sup .x y/ ! .y/ y 1 kŠ x2.˛;ˇ/ kD1 .l/ C kgkL1 .˛;ˇ/ f L1 .˛;ˇ/
Ckf kL1 .˛;ˇ/ g .l/
L1 .˛;ˇ/
# .ˇ ˛/l k!k1 : (1.49) .l 1/Š
.l/
(ii) Assume further that f; g; f ; g .l/ 2 L1 .˛; ˇ/ : Then "( k!k1 .f; g/ kf kL1 .˛;ˇ/ kgkL1 .˛;ˇ/ C kgkL1 .˛;ˇ/ kf kL1 .˛;ˇ/ 2 !) h l1 i.k/ X 1 k sup .x y/ ! .y/ y L .˛;ˇ/ kŠ x2.˛;ˇ/ 1 kD1 ( .l/ C kgkL1 .˛;ˇ/ f L1 .˛;ˇ/
Ckf kL1 .˛;ˇ/ g.l/
L1 .˛;ˇ/
.ˇ ˛/lC1 k!k1 .l 1/Š
)# :
(1.50) (iii) Let p; q > 1 W Then
1 p
C
k!k1 .f; g/ 2
1 q
"(
D 1I assume further that f; g; f
.l/
; g.l/ 2 Lp .˛; ˇ/ :
kgkL1 .˛;ˇ/ kf kLp .˛;ˇ/ C kf kL1 .˛;ˇ/ kgkLp .˛;ˇ/ !) h l1 i.k/ X 1 k sup .x y/ ! .y/ y L .˛;ˇ/ kŠ x2.˛;ˇ/ q kD1 ( .l/ C kgkL1 .˛;ˇ/ f Lp .˛;ˇ/
C kf kL1 .˛;ˇ/ g .l/
Lp .˛;ˇ/
lC1 p1
.ˇ ˛/ .l 1/Š
)#
k!k1
:
(1.51)
1.3
Main Results
17
Proof. By (1.43), (1.37) and by Theorems 1.23, 1.24, 1.25. Next, we give a series of Ostrowski-type inequalities. Theorem 1.28. LetR l 2 N; Œa; b R; a < ˛ < ˇ < b and ! 2 L1 .R/, sup p! Œ˛; ˇ, R ! .x/ dx D 1: Assume f on Œa; b W f .l1/ exists and is absolutely continuous on Œa; b : Then for any x 2 .˛; ˇ/ we get ˇ ˇ Z ˇ ˇ ˇ ˇ ˇ l1 f .y/ ! .y/ dy Q f .x/ˇ ˇf .x/ ˇ ˇ ˛ .l/ k!kL1 .R/ f L1 .˛;ˇ/ .ˇ ˛/l1 (1.52) WD A1 : .l 1/Š If additionally we assume k!kL1 .R/ < 1, then 8 x 2 .˛; ˇ/, we obtain ˇ ˇ Z ˇ ˇ ˇ ˇ ˇ f .y/ ! .y/ dy ˇ ˇf .x/ ˇ ˇ ˛ ! ! l1 X .ˇ x/kC1 C .x ˛/kC1 .k/ k!kL1 .R/ f 1 .k C 1/Š kD1 k!kL1 .R/ f .l/ L1 .˛;ˇ/ .ˇ ˛/l1 C (1.53) WD B1 .x/ : .l 1/Š Proof. By (1.41), (1.23), and (1.30). Theorem 1.29. All as in Theorem 1.28. Assume f 2 C l .Œa; b/. Then 8 x 2 .˛; ˇ/ ; ˇ ˇ Z ˇ ˇ ˇ ˇ ˇ l1 f .y/ ! .y/ dy Q f .x/ˇ ˇf .x/ ˇ ˇ ˛ k!kL1 .R/ f .l/ 1 .ˇ x/l C .x ˛/l DW A2 .x/ : (1.54) lŠ If additionally we assume k!kL1 .R/ < 1; then 8 x 2 .˛; ˇ/ ; ˇ ˇ Z ˇ ˇ ˇ ˇ ˇ f .y/ ! .y/ dy ˇ ˇf .x/ ˇ ˇ ˛ ! ! l1 X .ˇ x/kC1 C .x ˛/kC1 .k/ k!kL1 .R/ f 1 .k C 1/Š kD1 k!kL1 .R/ f .l/ 1 .ˇ x/l C .x ˛/l DW B2 .x/ : C lŠ
(1.55)
18
1 Univariate Integral Inequalities Based on Sobolev Representations
Proof. By (1.41), (1.24), and (1.30). We continue with
loc .a; b/ and rest as in Theorem 1.30. Let all as in Theorem 1.28 or f 2 W1l Theorem 1.28. Then 8 x 2 .˛; ˇ/ (or almost every x 2 .˛; ˇ/, respectively), we get ˇ ˇ Z ˇ ˇ ˇ ˇ ˇ l1 f .y/ ! .y/ dy Q f .x/ˇ E .f / .x/ WD ˇf .x/ ˇ ˇ ˛
.l/ k!kL1 .a;b/ f
L1 .˛;ˇ/
.ˇ ˛/l1 DW A3
.l 1/Š
(1.56)
Additionally, if k!kL1 .R/ < 1; 8 x 2 .˛; ˇ/ (or almost every x 2 .˛; ˇ/, respectively), we derive ˇ ˇ Z ˇ ˇ ˇ ˇ ˇ .f / .x/ WD ˇf .x/ f .y/ ! .y/ dy ˇ ˇ ˇ ˛
! l1 X .ˇ ˛/k .k/ k!kL1 .R/ f L1 .˛;ˇ/ kŠ kD1
C
.l/ k!kL1 .a;b/ f
L1 .˛;ˇ/
.ˇ ˛/l1
.l 1/Š
DW B3 :
(1.57)
Proof. By (1.41), (1.23), and (1.29).
loc Theorem 1.31. Let all as in Theorem 1.28 or f 2 W1l .a; b/ and rest as in .l/
2 L1 .˛; ˇ/ : Then 8 x 2 .˛; ˇ/ (or almost Theorem 1.28. Assume furtherf every x 2 .˛; ˇ/, respectively), we get
E .f / .x/
.l/ k!kL1 .R/ f
L1 .˛;ˇ/
lŠ
.ˇ x/l C .x ˛/l
DW A4 .x/ : (1.58)
1.3
Main Results
19
.k/
Additionally f 2 L1 .˛; ˇ/, k D 1; :::; l 1 and if k!kL1 .R/ < 1, then 8 x 2 .˛; ˇ/ (or almost every x 2 .˛; ˇ/, respectively), we get 0 .f / .x/ @
l1 X kD1
C
1 0 .ˇ x/kC1 C .x ˛/kC1 .k/ @ A f .k C 1/Š
1 A k!kL
1 .R/
L1 .˛;ˇ/
.l/ k!kL1 .R/ f
L1 .˛;ˇ/
.ˇ x/l C .x ˛/l
lŠ
DW B4 .x/ :
(1.59)
Proof. By (1.41), (1.24), and (1.30).
loc .a; b/ and rest as in Theorem 1.32. Let all as in Theorem 1.28 or f 2 W1l .l/
Theorem 1.28. Let p; q > 1 W p1 C q1 D 1: Assume further f 2 Lp .˛; ˇ/ : Then 8 x 2 .˛; ˇ/ (or almost every x 2 .˛; ˇ/, respectively), we get
E .f / .x/
.l/ k!kL1 .R/ f
Lp .˛;ˇ/
.l 1/Š .ˇ x/q.l1/C1 C .x ˛/q.l1/C1 q .l 1/ C 1
! q1 DW A5 .x/ : (1.60)
.k/
Additionally, if f 2 Lp .˛; ˇ/, k D 1; :::; l 1 and k!kL1 .R/ < 1, then 8 x 2 .˛; ˇ/ (or almost every x 2 .˛; ˇ/, respectively), we get .f / .x/ 0 B B @
l1 X kD1
1 1 q1 0 .k/ f .ˇ x/.kqC1/ C .x ˛/.kqC1/ Lp .˛;ˇ/ C C @ A A kq C 1 kŠ
k!kL1 .R/ C A5 .x/ DW B5 .x/ : Proof. By (1.41), (1.25), and (1.31).
(1.61)
20
1 Univariate Integral Inequalities Based on Sobolev Representations
We further give Theorem 1.33. Let all as in Theorem 1.31. Here assume ! 2 L1 .R/. Then 8 x 2 .˛; ˇ/ (or almost every x 2 .˛; ˇ/, respectively), we get ˇ ˇ Z ˇ ˇ ˇ ˇ ˇ f .y/ ! .y/ dy ˇ .f / .x/ WD ˇf .x/ ˇ ˇ ˛ 0 1 .k/ l1 f .ˇ ˛/k X L1 .˛;ˇ/ B C @ A k!kL1 .R/ C A4 .x/ kŠ kD1
DW B6 .x/ :
(1.62)
Proof. By (1.32) and (1.58). Theorem 1.34. Let all be as in Theorem 1.32. Here assume ! 2 Lq .˛; ˇ/ ; q > 1: Then 8 x 2 .˛; ˇ/ (or almost every x 2 .˛; ˇ/, respectively), it holds ! l1 X .ˇ ˛/k .k/ .f / .x/ k!kLq .˛;ˇ/ C A5 .x/ DW B7 .x/ : f Lp .˛;ˇ/ kŠ kD1 (1.63) Proof. By (1.33) and (1.60).
loc .a; b/ and rest as in TheTheorem 1.35. Let all as in Theorem 1.28 or f 2 W1l .k/
orem 1.28. Let p; q; r > 1 W p1 C q1 C 1r D 1; f 2 Lp .˛; ˇ/, k D 1; :::; l 1I ! 2 Lq .˛; ˇ/ : Then 8 x 2 .˛; ˇ/ (or almost every x 2 .˛; ˇ/, respectively), it holds ˇ ˇ Z ˇ ˇ ˇ ˇ ˇ f .y/ ! .y/ dy R0;l f .x/ˇ ˇf .x/ ˇ ˇ ˛ .k/ 1 0 ! 1r f l1 .krC1/ .krC1/ .ˇ x/ C .x ˛/ Lp .˛;ˇ/ C BX @ A k!kLq .˛;ˇ/ kŠ .kr C 1/ kD1
DW ˚ .x/ :
(1.64)
Proof. By (1.34) and (1.41). We also give .l2/ .R/, sup p! Œ˛; ˇ, RTheorem 1.36. Let N.l2/3 l > 1 and ! 2 C ! .x/ dx D 1; ! is absolutely continuous on Œa; b ; Œ˛; ˇ .a; b/ I R loc .a; b/ : For every x 2 .˛; ˇ/ a; b 2 R: Here f 2 C l .Œa; b/ or f 2 W1l (or almost every x 2 .˛; ˇ/, respectively), we get for ˇ ˇ Z ˇ ˇ ˇ ˇ ˇ .f / .x/ WD ˇf .x/ f .y/ ! .y/ dy ˇ ˇ ˇ ˛
1.3
Main Results
21
that: (i) It holds ! h l1 i.k/ X 1 k .x y/ ! .y/ .f / .x/ kf kL1 .˛;ˇ/ y kŠ 1 kD1 .l/ .ˇ ˛/l1 k!kL1 .R/ f L1 .˛;ˇ/ C DW C1 .x/ : (1.65) .l 1/Š (ii) If f; f
.l/
2 L1 .˛; ˇ/ ; then
! h l1 i.k/ X 1 k .x y/ ! .y/ .f / .x/ kf kL1 .˛;ˇ/ y L .˛;ˇ/ kŠ 1 kD1 .l/ .ˇ x/l C .x ˛/l k!kL1 .R/ f L1 .˛;ˇ/ C lŠ (1.66) DW C2 .x/ : (iii) Let p; q > 1 W
1 p
C
1 q
D 1: Assume further that f; f
.l/
2 Lp .˛; ˇ/ : Then
! h l1 i.k/ X 1 k .f / .x/ .x y/ ! .y/ kf kLp .˛;ˇ/ y kŠ Lq .˛;ˇ/ kD1 .l/ k!kL1 .R/ f Lp .˛;ˇ/ C .l 1/Š ! q1 .ˇ x/q.l1/C1 C .x ˛/q.l1/C1 q .l 1/ C 1 DW C3 .x/ :
(1.67)
Proof. By (1.37) and Theorems 1.30–1.32. We finish Ostrowski-type inequalities with m 2 N; m < lI ! 2 C .l2/ .R/, sup p! Œ˛; ˇ, RTheorem 1.37. Let l;.l2/ is absolutely continuous on Œa; b ; Œ˛; ˇ .a; b/ I R ! .x/ dx D 1; ! loc l .a; b/ : For every x 2 .˛; ˇ/ a; b 2 R: Here f 2 C .Œa; b/ or f 2 W1l (or almost every x 2 .˛; ˇ/, respectively), we get for ˇ ˇ Z ˇ ˇ .m/ ˇ ˇ ˇ m .m/ l1 .x/ .1/ f .y/ ! .y/ dy Qm f .x/ˇ ; (1.68) Eˇ .f / .x/ WD ˇf ˇ ˇ ˛
22
1 Univariate Integral Inequalities Based on Sobolev Representations
and
ˇ ˇ Z ˇ ˇ .m/ ˇ ˇ ˇ m .m/ ˇ .f / .x/ WD ˇf .x/ .1/ f .y/ ! .y/ dy ˇ ˇ ˇ ˛
(1.69)
that (i) It holds .l/ k!kL1 .R/ f
Eˇ .f / .x/
L1 .˛;ˇ/
.ˇ ˛/lm1 DW E1 ;
.l m 1/Š
(1.70)
and lm1 X
ˇ .f / .x/
kD1
! h i.kCm/ 1 .x y/k ! .y/ kf kL1 .˛;ˇ/ y kŠ 1
CE1 DW G1 .x/ : (ii) If f
.l/
(1.71)
2 L1 .˛; ˇ/ ; then .l/ k!kL1 .R/ f
L1 .˛;ˇ/
Eˇ .f / .x/
.l m/Š
.ˇ x/lm C .x ˛/lm DW E2 .x/ ;
(1.72)
if additionally we assume f 2 L1 .˛; ˇ/ ; then ˇ .f / .x/
lm1 X kD1
! h i.k/ 1 .x y/k ! .y/ y L .˛;ˇ/ kŠ 1
kf kL1 .˛;ˇ/ C E2 .x/ DW G2 .x/ : (iii) Let p; q > 1 W
1 p
C
1 q
Eˇ .f / .x/
D 1; assume further that f
.l/
(1.73)
2 Lp .˛; ˇ/ ; then
.l/ k!kL1 .R/ f
Lp .˛;ˇ/
.l m 1/Š .ˇ x/q.lm1/C1 C .x ˛/q.lm1/C1 q .l m 1/ C 1
DW E3 .x/ ;
! q1
(1.74)
1.3
Main Results
23
and if additionally f 2 Lp .˛; ˇ/ ; then ˇ .f / .x/
lm1 X kD1
! h i.kCm/ 1 .x y/k ! .y/ kf kLp .˛;ˇ/ y kŠ Lq .˛;ˇ/
CE3 .x/ DW G3 .x/ :
(1.75)
Proof. By (1.23)–(1.25), (1.40), and (1.42). We make Remark 1.38. In preparation to present comparison of integral means inequalities, we consider .˛ 1 ; ˇ 1 / .˛; ˇ/. We consider also a weight function R 0 which is Lebesgue integrable on R with sup p Œ˛1 ; ˇ 1 Œa; b, and R .x/ dx D 1. Rˇ Clearly here ˛11 .x/ dx D 1: 1 For example, for x 2 .˛ 1 ; ˇ 1 /, .x/ WD ˇ ˛ , zero elsewhere, etc. 1 1 We will apply the following principle: In general a constraint of the form jF .x/ Gj ", where F is a function and G; " real numbers so that all make sense, implies that ˇ ˇZ ˇ ˇ ˇ F .x/ .x/ dx G ˇ ": (1.76) ˇ ˇ R
Next we present a series of comparison of integral means inequalities based on Ostrowski-type inequalities given in this chapter. We use Remark 1.38. Theorem 1.39. All as in Theorem 1.28. Then ˇZ Z ˇ Z ˇ1 ˇ ˇ1 ˇ f .x/ .x/ dx f .y/ ! .y/ dy Ql1 f .x/ u .f / WD ˇ ˇ ˛1 ˛ ˛1 A1 ;
ˇ ˇ ˇ .x/ dx ˇ ˇ (1.77)
and ˇZ ˇ ˇ1 ˇ m .f / WD ˇ f .x/ ˇ ˛1
ˇ ˇ ˇ .x/ dx f .y/ ! .y/ dy ˇ ˇ ˛ ! l1 X .ˇ ˛/kC1 .k/ k!kL1 .R/ f 1 .k C 1/Š kD1 k!kL1 .R/ f .l/ L1 .˛;ˇ/ .ˇ ˛/l1 : C .l 1/Š Z
ˇ
(1.78)
24
1 Univariate Integral Inequalities Based on Sobolev Representations
Proof. By Remark 1.38, Theorem 1.28, and the fact that the functions .ˇ x/kC1 C .x ˛/kC1 , k D 1; :::; l 1 are positive and convex with maximum .ˇ ˛/kC1 : Theorem 1.40. All as in Theorem 1.29. Then k!kL1 .R/ f .l/ 1 .ˇ ˛/l ; u .f / lŠ
(1.79)
and ! l1 X .ˇ ˛/kC1 .k/ m .f / k!kL1 .R/ f 1 .k C 1/Š kD1
k!kL1 .R/ f .l/ 1 .ˇ ˛/l C : lŠ
(1.80)
Proof. Just maximize A2 .x/ of (1.54) and B2 .x/ of (1.55), etc.
Theorem 1.41. All as in Theorem 1.30. Then u .f / A3 ;
(1.81)
m .f / B3 :
(1.82)
and Theorem 1.42. All as in Theorem 1.31. Then
u .f /
.l/ k!kL1 .R/ f
L1 .˛;ˇ/
.ˇ ˛/l ;
lŠ
(1.83)
and ! l1 X .ˇ ˛/kC1 .k/ m .f / k!kL1 .R/ f L1 .˛;ˇ/ .k C 1/Š kD1
C
.l/ k!kL1 .R/ f
L1 .˛;ˇ/
.ˇ ˛/l
lŠ
:
(1.84)
Theorem 1.43. All as in Theorem 1.32. Then Z u .f /
ˇ1 ˛1
A5 .x/
.x/ dx;
(1.85)
1.3
Main Results
25
and
Z
ˇ1
m .f /
˛1
B5 .x/
.x/ dx:
(1.86)
ˇ ˇR Proof. By the principle: if jF .x/ Gj " .x/, then ˇ F .x/ .x/ dx G ˇ R " .x/ .x/ dx; etc: Here A5 .x/ as in (1.60) and B5 .x/ as in (1.61). Theorem 1.44. All as in Theorem 1.33. Then Z m .f /
ˇ1
˛1
B6 .x/
.x/ dx;
(1.87)
.x/ dx;
(1.88)
where B6 .x/ as in (1.62). Theorem 1.45. All as in Theorem 1.34. Then Z m .f /
ˇ1
˛1
B7 .x/
where B7 .x/ as in (1.63). Theorem 1.46. All as in Theorem 1.35. Then ˇZ ˇ ˇ1 ˇ f .x/ ˇ ˇ ˛1 Z
ˇ1
Z
Z
ˇ
.x/ dx
f .y/ ! .y/ dy ˛
˚ .x/
ˇ1 ˛1
.R0;l f .x//
ˇ ˇ ˇ .x/ dx ˇ ˇ
.x/ dx;
(1.89)
˛1
where ˚ .x/ as in (1.64). We continue with Theorem 1.47. All as in Theorem 1.36. Then (i) 0 l1 X 1 m .f / @ sup kŠ x2Œ˛ 1 ;ˇ kD1
C
1
1 h i.k/ .x y/k ! .y/ A kf k .˛;ˇ/ L1 y 1
.l/ k!kL1 .R/ f
L1 .˛;ˇ/
.l 1/Š
.ˇ ˛/l1 :
(1.90)
26
1 Univariate Integral Inequalities Based on Sobolev Representations
(ii) If f; f
.l/
2 L1 .˛; ˇ/ ; then Z
ˇ1
m .f /
˛1
(iii) Let p; q > 1 W
1 p
C
1 q
C2 .x/
.x/ dx:
D 1I assume further f; f Z m .f /
ˇ1
˛1
C3 .x/
.l/
(1.91)
2 Lp .˛; ˇ/ ; then
.x/ dx:
(1.92)
Here C2 .x/ as in (1.66) and C3 .x/ as in (1.67). We finish the results about comparison of integral means with Theorem 1.48. All as in Theorem 1.37. Denote by ˇZ ˇ ˇ1 .m/ ˇ f .x/ um .f / WD ˇ ˇ ˛1 Z
ˇ1
˛1
.x/ dx .1/m
Z
ˇ
f .y/ ! .m/ .y/ dy
˛
l1 f .x/ Qm
ˇ ˇ ˇ .x/ dx ˇ ; ˇ
(1.93)
and ˇZ ˇ ˇ1 .m/ ˇ m .f / WD ˇ f .x/ ˇ ˛1
m
Z
ˇ
.x/ dx .1/
f .y/ !
.m/
˛
ˇ ˇ ˇ .y/ dy ˇ : ˇ
(1.94)
(i) It holds um .f / E1 ;
(1.95)
where E1 as in (1.70), and 0
lm1 X
m .f / @
kD1
1 sup kŠ x2Œ˛1 ;ˇ
1
1 h i.kCm/ A kf k .˛;ˇ/ C E1 ; .x y/k ! .y/ L1 y 1
(1.96) (ii) If f
.l/
2 L1 .˛; ˇ/ ; then um .f /
.l/ k!kL1 .R/ f .l m/Š
L1 .˛;ˇ/
.ˇ ˛/lm ;
(1.97)
1.3
Main Results
27
and if additionally assume f 2 L1 .˛; ˇ/ ; then 0
h i.kCm/ .x y/k ! .y/ y
lm1 X
m .f / @
C
1 sup kŠ x2Œ˛1 ;ˇ kD1 1 .l/ k!kL1 .R/ f
(iii) Let p; q > 1 W
L1 .˛;ˇ/
L1 .˛;ˇ/
C
1 q
;
D 1; assume further f Z um .f /
ˇ1
˛1
A kf kL
1 .˛;ˇ/
.ˇ ˛/lm
.l m/Š 1 p
1
E3 .x/
.l/
(1.98)
2 Lp .˛; ˇ/, then .x/ dx;
(1.99)
where E3 .x/ as in (1.74), and if additionally f 2 Lp .˛; ˇ/, then Z m .f /
ˇ1 ˛1
G3 .x/
.x/ dx;
(1.100)
where G3 .x/ as in (1.75). We need Remark 1.49 (Background). Let f be a convex function from .0; C1/ into R which is strictly convex at 1 with f .1/ D 0. Let .X; A; / be a measure space, where is a finite or a -finite measure on .X; A/. And let 1 ; 2 be two probability measures on .X; A/ such that 1 , 2 (absolutely continuous), e.g., D 1 C 2 : 1 2 , q D d the (densities) Radon-Nikodym derivatives of 1 , Denote by p D d d d 2 with respect to . Here we suppose that 0<˛
p ˇ, a.e. on X and ˛ 1 ˇ: q
The quantity
Z
f . 1 ; 2 / D
q .x/ f X
p .x/ d .x/ ; q .x/
(1.101)
was introduced by I. Csiszar in 1967, see [9], and is called f -divergence of the probability measures 1 and 2 . By Lemma 1.1 of [9], the integral (1.101) is welldefined and f . 1 ; 2 / 0 with equality only when 1 D 2 : Furthermore
f . 1 ; 2 / does not depend on the choice of . The concept of f -divergence was introduced first in [8] as a generalization of Kullback’s “information for
28
1 Univariate Integral Inequalities Based on Sobolev Representations
discrimination” or I -divergence (generalized entropy) [12, 13] and of R´enyi’s “information gain” (I -divergence of order ı) [15]. In fact the I -divergence of order 1 equals u log2 u . 1 ; 2 /. The choice f .x/ D .u 1/2 produces again a known measure of difference of distributions Rthat is called 2 -divergence. Of course the total variation distance j 1 2 j D X jp .x/ q .x/j d .x/ is equal to ju1j . 1 ; 2 /. Here by supposing f .1/ D 0 we can consider f . 1 ; 2 /, the f -divergence as a measure of the difference between the probability measures 1 ; 2 . The f -divergence is in general asymmetric in 1 and 2 . But since f is convex and strictly convex at 1 so is 1 (1.102) f .u/ D uf u and as in [9] we obtain
f . 2 ; 1 / D f . 1 ; 2 /:
(1.103)
In information theory and statistics many other divergences are used which are special cases of the above general Csiszar f -divergence, e.g., the Hellinger distance DH , ˛-distance D˛ , Bhattacharyya distance DB , Harmonic distance DHa , Jeffrey’s distance DJ , triangular discrimination D , for all these see, e.g., [5, 10]. The problem of finding and estimating the proper distance (or difference or discrimination) of two probability distributions is one of the major ones in Probability Theory. Here we provide a general probabilistic representation formula for f . 1 ; 2 /. Then we present tight estimates for the remainder involving a variety of norms of the engaged functions. Also are implied some direct general approximations for the Csiszar’s f -divergence. We give some applications. We make Remark 1.50. Here 0 < a < ˛ p.x/ q.x/ ˇ < b < C1, a e. on X and ˛ 1 ˇ. .l1/ exists and is absolutely continuous on Œa; b, l 2 N. Also suppose that f Furthermore f is convex from .0; C1/ into R, strictly convex at 1 with f .1/ D 0. R Let ! 2 L1 .R/, sup p! Œ˛; ˇ, R ! .x/ dx D 1. Then 8 x 2 .˛; ˇ/ we get by Theorem 1.10, as in (1.41), that Z
ˇ
f .x/ D ˛
f .y/ ! .y/ dy C Ql1 f .x/ C R0;l f .x/:
Therefore f
p .x/ q .x/
a.e. on X:
Z
ˇ
D ˛
f .y/ ! .y/ dy C Q
l1
f
p .x/ q .x/
C R0;l f
p .x/ ; q .x/
1.3
Main Results
29
Hence q .x/ f
p .x/ q .x/
Z D q .x/
ˇ
f .y/ ! .y/ dy
˛
C q .x/ Q
l1
f
p .x/ q .x/
C q .x/ R0;l f
p .x/ ; q .x/
a.e. on X . Therefore we get the representation of f -divergence of 1 and 2 ,
Z ˇ p .x/ q .x/ f f .y/ ! .y/ dy d .x/ D
f . 1 ; 2 / D q .x/ ˛ X Z p .x/ C q .x/ Q l1 f d .x/ q .x/ X Z p .x/ q .x/ R0;l f d .x/ : (1.104) C q .x/ X Z
Z
Call Q WD
X
Z
and
q .x/ Ql1 f
R WD
X
q .x/ R0;l f
p .x/ d .x/ ; q .x/
(1.105)
p .x/ d .x/ : q .x/
(1.106)
We estimate Q and R . If k!kL1 .R/ < 1, we get by (1.29) that ! l1 X .ˇ ˛/k .k/ k!kL1 .R/ : jQ j f L1 .˛;ˇ/ kŠ
(1.107)
kD1
Notice if l D 1, then always Q D 0. Next if again k!kL1 .R/ < 1; then (by (1.30)) 0 B jQ j @
Z X
0
l1 ˇ X B q .x/ @
p.x/ q.x/
L1 .˛;ˇ/
C
p.x/ q.x/
˛
kC1
.k C 1/Š
kD1
f .k/
kC1
1
1
C C A d .x/A k!kL1 .R/ :
(1.108)
30
1 Univariate Integral Inequalities Based on Sobolev Representations 1 p
Let now p; q > 1 W 0 B jQ j B @
C
1 q
D 1 and again k!kL1 .R/ < 1. Then (by (1.31)) 0
Z X
0 l1 ˇ X B B q .x/ B @ @
p.x/ q.x/
.kqC1/
1 p .˛;ˇ/
kŠ
p.x/ q.x/
˛
.kqC1/ 1 q1
kq C 1
kD1
.k/ f L
C
C A
1
C C C d .x/ C k!k .R/ : L1 A A
(1.109)
Next assume ! 2 L1 .R/, then (by (1.32)) l1 f .k/ X L
1 .˛;ˇ/
jQ j
kŠ
kD1
If p; q > 1 W
1 p
C
1 q
.ˇ ˛/k
! k!kL1 .R/ :
(1.110)
D 1 and ! 2 Lq .˛; ˇ/, then (by (1.33))
! l1 X .ˇ ˛/k .k/ jQ j k!kLq .˛;ˇ/ : f Lp .˛;ˇ/ kŠ
(1.111)
kD1
Assume p; q; r > 1 W 0 B jQ j @
Z
1 p
C
1 q
C
1 r
D 1 and ! 2 Lq .˛; ˇ/, then (by (1.34))
0
l1 .k/ Lp .˛;ˇ/ BX f q .x/ @ kŠ X kD1
0 ˇ B @
p.x/ q.x/
.krC1/
C
p.x/ q.x/
kr C 1
1 .krC1/ 1 1r 1 ˛ C C C C A C A d .x/A
k!kLq .˛;ˇ/ :
(1.112)
We make Remark 1.51 (continuation of Remark 1.50). Here l > 1, ! 2 C .l2/ .R/, R sup p! Œ˛; ˇ, R ! .x/ dx D 1, and ! .l2/ is absolutely continuous on Œa; b : Then (by (1.35)) 0
Z Q D
q .x/ @ X
Z l1 X .1/k kD1
kŠ
˛
ˇ
1 0" #.k/ 1 k @ p .x/ y ! .y/ A f .y/ dy A d .x/ : q .x/ y
(1.113)
1.3
Main Results
31
Hence by (1.37) we obtain jQ j min of 0 80 l1 ˆ X R ˆ ˆ 1 @ @ ˆ q .x/ ˆ X kŠ ˆ ˆ ˆ ˆ 0kD1 ˆ ˆ0 ˆ l1 ˆ X R ˆ < @ q .x/ @ 1 X
kŠ
1 1
.k/ k p.x/ A d .x/A kf kL1 .˛;ˇ/ , q.x/ y ! .y/ y 1 1 1
.k/ k p.x/ A d .x/A kf kL .˛;ˇ/ ; q.x/ y ! .y/ 1 y
kD1 L1 .˛;ˇ/ ˆ ˆ ˆ 1 1 ˆ when p; q > 1 W C D 1, we have ˆ p ˆ0 1 0 1 ˆ q ˆ .k/ ˆ l1 k ˆ X R ˆ p.x/ 1 ˆ A d .x/A kf kL .˛;ˇ/ : @ q .x/ @ ˆ y ! .y/ ˆ kŠ p : X q.x/ y Lq .˛;ˇ/
kD1
(1.114) We also make Remark 1.52 (another continuation of Remark 1.50). Here we estimate the remainder R of (1.104). By (1.23) and (1.106), we obtain jR j
k!kL1 .a;b/ f .l/ L1 .˛;ˇ/ .ˇ ˛/l1 .l 1/Š
:
If f .l/ 2 L1 .˛; ˇ/, then (by (1.24)) we obtain k!kL1 .a;b/ f .l/ L1 .˛;ˇ/ Z p .x/ l q .x/ ˇ jR j lŠ q .x/ X ! ! l p .x/ C d .x/ : ˛ q .x/ Let now p; q > 1 W
jR j
1 p
C
1 q
(1.116)
D 1: Here f .l/ 2 Lp .˛; ˇ/ ; then (by (1.25)) we get
k!kL1 .a;b/ f .l/ Lp .˛;ˇ/
q .x/
1
p .x/ C ˛ q .x/
.q.l1/C1/
p .x/ .q.l1/C1/ ˇ q .x/
Z
.q .l 1/ C 1/ q .l 1/Š
(1.115)
X
! q1
1 d .x/A :
(1.117)
Finally we observe that Z
f . 1 ; 2 /
˛
ˇ
f .y/ ! .y/ dy D Q C R ;
(1.118)
32
1 Univariate Integral Inequalities Based on Sobolev Representations
and ˇ ˇ Z ˇ ˇ ˇ ˇ ˇ T WD ˇ f . 1 ; 2 / f .y/ ! .y/ dy ˇ jQ j C jR j : ˇ ˇ ˛
(1.119)
Then one by the above estimates of jQ j and jR j can estimate T , in a number of cases.
1.4 Applications Example 1.53. Let V WD fx 2 R W jx x0 j < g, x0 2 R, and ' .x/ WD
8 < :
1
e 0,
.xx0 /2
1
2
; if jx x0 j < ; if jx x0 j :
(1.120)
R Call c WD R ' .x/ dx > 0, then ˚ .x/ WD 1c ' .x/ 2 C01 .R/ (space of continuously infinitely R 1 many times differentiable functions of compact support) with sup p˚ D V and 1 ˚ .x/ dx D 1 and max j˚j cons tan t 1 : We call ˚ a cut-off function. 1 One for this chapter’s results by choosing ! .x/ D ˚ .x/ or ! .x/ D 2 , etc., can give lots of applications. Due to lack of space we avoid it. Instead, selectively, we give some special cases inequalities. We start with Chebyshev–Gr¨uss-type inequalities.
Corollary 1.54 (to Theorem 1.22). Let f; g 2R C 1 .Œa; b/, Œa; b R, .˛; ˇ/ .a; b/. Let also ! 2 L1 .R/, sup p! Œ˛; ˇ, R ! .x/ dx D 1: Then ˇZ ! Z !ˇ Z ˇ ˇ ˇ ˇ ˇ ˇ ˇ ! .x/ f .x/ g .x/ dx ! .x/ f .x/ dx ! .x/ g .x/ dx ˇ ˇ ˇ ˇ ˛ ˛ ˛ .ˇ ˛/2 k!kL1 .R/ k!k1;.˛;ˇ/ 2 0 kgk1;.˛;ˇ/ f 1;.˛;ˇ/ C kf k1;.˛;ˇ/ g 0 1;.˛;ˇ/ :
(1.121)
If f D g, then ˇ !2 ˇˇ ˇZ ˇ Z ˇ ˇ ˇ ˇ ! .x/ f 2 .x/ dx ! .x/ f .x/ dx ˇˇ ˇ ˛ ˇ ˛ ˇ k!kL1 .R/ k!k1;.˛;ˇ/ .ˇ ˛/2 kf k1;.˛;ˇ/ f 0 1;.˛;ˇ/ :
(1.122)
1.4
Applications
33
loc Corollary 1.55 (to Theorem 1.23). Let f 2 W11 .a; b/ I a; b 2 RI .˛; ˇ/ 1 for x 2 Œ˛; ˇ, and zero elsewhere. Then .a; b/ ; ! .x/ WD ˇ˛ ˇ ˇ Z ˇ ˇ 1 1 ˇ f 2 .x/ dx ˇˇ ˛ .ˇ ˛/2 ˛ ˇ
Z
ˇ ˛
!2 ˇˇ ˇ kf kL1 .˛;ˇ/ fw.1/ L1 .˛;ˇ/ f .x/ dx ˇˇ : .ˇ ˛/ ˇ (1.123)
We continue with an Ostrowski-type inequality. Corollary 1.56 (to Theorem 1.30). All as in Theorem 1.30. Case of l D 1. Then, for any x 2 .˛; ˇ/ (or for almost every x 2 .˛; ˇ/, respectively), we get ˇ ˇ Z ˇ ˇ 0 ˇ ˇ ˇ f .y/ ! .y/ dy ˇ k!kL1 .R/ f : ˇf .x/ ˇ ˇ L1 .˛;ˇ/ ˛
(1.124)
Next comes a comparison of means inequality. Corollary 1.57. All here as in Corollary 1.56 and Remark 1.38. Then ˇZ ˇ ˇ1 ˇ f .x/ ˇ ˇ ˛1
Z
ˇ
.x/ dx ˛
ˇ ˇ 0 ˇ f .y/ ! .y/ dy ˇ k!kL1 .R/ f : ˇ L1 .˛;ˇ/
(1.125)
Proof. By (1.124). We finish with an application of f -divergence. Remark 1.58. All here as in Background 1.49 and Remark 1.50. Case of l D 1. By (1.104) we get Z
f . 1 ; 2 / D
ˇ
Z f .y/ ! .y/ dy C
˛
X
That is here
Z R D
q .x/ R0;1 f
X
q .x/ R0;1 f
p .x/ d .x/ : q .x/
p .x/ d .x/ : q .x/
(1.126)
(1.127)
By (1.115) here we get that jR j k!kL1 .a;b/ f 0 L1 .˛;ˇ/ :
(1.128)
If f 0 2 L1 .˛; ˇ/, then here we get jR j k!kL1 .a;b/ f 0 L1 .˛;ˇ/ .ˇ ˛/ :
(1.129)
34
1 Univariate Integral Inequalities Based on Sobolev Representations
Let now p; q > 1 W
1 p
C
1 q
D 1 and assume f 0 2 Lp .˛; ˇ/, then here we obtain
1 jR j k!kL1 .a;b/ f 0 Lp .˛;ˇ/ .ˇ ˛/ q :
(1.130)
Also notice here that Z K WD f . 1 ; 2 /
ˇ ˛
f .y/ ! .y/ dy D R ;
(1.131)
(l D 1 case). So the estimates (1.128)–(1.130) are also estimates for K:
References 1. G.A. Anastassiou, Quantitative Approximations, Chapman & Hall/CRC, Boca Raton, New York, 2000. 2. G.A. Anastassiou, Probabilistic Inequalities, World Scientific, Singapore, New Jersey, 2010. 3. G.A. Anastassiou, Advanced Inequalities, World Scientific, Singapore, New Jersey, 2011. 4. G.A. Anastassiou, Univariate Inequalities based on Sobolev Representations, Studia Mathematica-Babes Bolyai, accepted 2011. 5. N.S. Barnett, P. Cerone, S.S. Dragomir, A. Sofo, Approximating Csiszar’s f -divergence by the use of Taylor’s formula with integral remainder, (paper #10, pp. 16), Inequalities for Csiszar’s f -Divergence in Information Theory, S.S. Dragomir (ed.), Victoria University, Melbourne, Australia, 2000. On line: http://rgmia.vu.edu.au 6. V. Burenkov, Sobolev spaces and domains, B.G. Teubner, Stuttgart, Leipzig, 1998. 7. P.L. Chebyshev, Sur les expressions approximatives des integrales definies par les autres prises entre les mˆemes limites, Proc. Math. Soc. Charkov, 2(1882), 93-98. 8. I. Csiszar, Eine Informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizit¨at von Markoffschen Ketten, Magyar Tud. Akad. Mat. Kutato Int. K¨ozl. 8 (1963), 85-108. 9. I. Csiszar, Information-type measures of difference of probability distributions and indirect observations, Studia Math. Hungarica 2 (1967), 299-318. 10. S.S. Dragomir (ed.), Inequalities for Csiszar f -Divergence in Information Theory, Victoria University, Melbourne, Australia, 2000. On-line: http://rgmia.vu.edu.au h Rb 1 ¨ .x/ g .x/ dx 11. G. Gr¨uss, Uber das Maximum des absoluten Betrages von a f ba i R R b b 1 .ba/ , Math. Z. 39 (1935), pp. 215-226. 2 a f .x/ dx a g .x/ dx 12. S. Kullback, Information Theory and Statistics, Wiley, New York, 1959. 13. S. Kullback, R. Leibler, On information and sufficiency, Ann. Math. Statist., 22 (1951), 79-86. 14. A. Ostrowski, Uber die Absolutabweichung einer differentiabaren Funcktion von ihrem Integralmittelwert, Comment. Math. Helv. 10(1938), 226-227. 15. A. R´enyi, On measures of entropy and information, Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, I, Berkeley, CA, 1960, 547-561.
Chapter 2
Multivariate Integral Inequalities Deriving from Sobolev Representations
Here we present very general multivariate tight integral inequalities of Chebyshev–Gr¨uss, Ostrowski types and of comparison of integral means. These rely on the well-known Sobolev integral representation of a function. The inequalities engage ordinary and weak partial derivatives of the involved functions. We give also applications. On the way to prove the main results we obtain important estimates for the averaged Taylor polynomials and remainders of Sobolev integral representations. The exposed results are thoroughly discussed. This chapter relies on [4].
2.1 Introduction This chapter is greatly motivated by the following theorems: Theorem A (Chebychev, 1882, [7]). Let f; g W Œa; b ! R absolutely continuous functions. If f 0 ; g0 2 L1 .Œa; b/, then ˇ ! Z !ˇ Z b ˇ ˇ 1 Z b b 1 ˇ ˇ f .x/ g .x/ dx f .x/ dx g .x/ dx ˇ ˇ 2 ˇ ˇb a a .b a/ a a
1 .b a/2 f 0 1 g 0 1 : 12
Theorem B (G. Gruss, ¨ 1935, [8]). Let f; g integrable functions from Œa; b ! R, such that m f .x/ M , g .x/ , for all x 2 Œa; b ; where m; M; ; 2 R. Then ˇ ! Z !ˇ Z b ˇ ˇ 1 Z b b 1 ˇ ˇ f .x/ g .x/ dx f .x/ dx g .x/ dx ˇ ˇ ˇ ˇb a a .b a/2 a a
1 .M m/ . / : 4
In 1938, A. Ostrowski [9] proved. G.A. Anastassiou, Inequalities Based on Sobolev Representations, SpringerBriefs in Mathematics 2, DOI 10.1007/978-1-4614-0201-5 2, © George A. Anastassiou
35
36
2 Multivariate Inequalities Based on Sobolev Representations
Theorem C. Let f W Œa; b ! R be continuous on Œa; b and differentiable on .a; b/ whose derivative f 0 W .a; b/ ! R is bounded on .a; b/, i.e., kf 0 k1 D sup jf 0 .t/j < C1: Then t 2.a;b/
ˇ ˇ " 2 # ˇ 1 Z b ˇ x aCb 1 ˇ ˇ 2 f .t/ dt f .x/ˇ .b a/ f 0 1 ; C ˇ 2 ˇb a a ˇ 4 .b a/ for any x 2 Œa; b. The constant
1 4
is the best possible.
See also [1–3] for related works that inspired as well this chapter. In this chapter using the Sobolev-type representation formulae, see Theorems 2.6, 2.8, 2.11 and 2.23, also Corollaries 2.12 and 2.13, we estimate first their remainders and then the involved averaged Taylor polynomials. Based on these estimates we establish lots of very tight inequalities on Rn , n 2 N, of Chebyshev–Gr¨uss type, Ostrowski type and of Comparison of integral means with applications. The results involve ordinary and weak partial derivatives and they go to all possible directions using various norms. All of our machinery comes from the excellent monograph by V. Burenkov, [6].
2.2 Background Here we follow [6]. For a measurable nonempty set ˝ Rn , n 2 N we shall denote by Lloc p .˝/ (1 p 1) - the set of functions defined on ˝ such that for each compact K ˝, f 2 Lp .K/. Definition 2.1. Let ˝ Rn be an open set, ˛ 2 ZnC , ˛ ¤ 0 and f; g 2 Lloc 1 .˝/. The function g is a weak derivative of the function f of order ˛ on ˝ (briefly g D Dw˛ f ) if 8 ' 2 C01 .˝/ (i.e., ' 2 C 1 .˝/ compactly supported in ˝) Z Z fD ˛ 'dx D .1/j˛j g' dx: (2.1) ˝
˝
Wpl
Definition 2.2. .˝/ (l 2 N, 1 p 1) – Sobolev space, which is the Banach space of functions f 2 Lp .˝/ such that 8 ˛ 2 ZnC where j˛j l the weak derivatives Dw˛ f exist on ˝ and Dw˛ f 2 Lp .˝/, with the norm X D ˛ f : (2.2) kf kWpl .˝/ D w L .˝/ j˛jl
p
.loc/ .˝/ WD Definition 2.3. For l 2 N, we define the Sobolev-type local space W1l ff W ˝ ! R W f 2 L1loc .˝/ and all f -distributional partials of orders l belong to L1loc .˝/g D ff 2 Lloc 1 .˝/ W for each open set G compactly embedded into ˝, f 2 W1l .G/g:
2.2
Background
37
Definition 2.4. A domain ˝ Rn is called star-shaped with respect to the point y 2 ˝ if 8 x 2 ˝ the closed interval line segment Œx; y ˝. A domain ˝ Rn is called star-shaped with respect to an open ball B ˝ if 8 y 2 B and 8 x 2 ˝ we have Œx; y ˝. We call the set Vx D Vx;B D [y2B .x; y/ D convex hull of fxg [ B; a conic body with vertex x constructed on the open ball B (if x 2 B, then Vx D B; if B ˝ and x 2 B, then Vx D B). In fact Vx for otherwise is the region consisting of B and the part of the cone with vertex at x, tangent to the sphere of B, which lies between x and B: Next comes the multidimensional Taylor’s formula. Theorem 2.5. Let ˝ Rn be a domain star-shaped with respect to the point x0 2 ˝, l 2 N and f 2 C l .˝/. Then 8 x 2 ˝ f .x/ D
X X .D ˛ f / .x0 / .x x0 /˛ C l ˛Š
j˛j
.x x0 /˛ ˛Š
Z 0
j˛jDl
1
.1 t/l1 .D ˛ f / .x0 C t .x x0 // dt
(2.3)
(here we mean x0 Ct .x xP 0 / D .x01 C t .x1 x01 / ; : : : ; x0n C t .xn x0n //, ˛ D .˛ 1 ; : : : ; ˛ n / 2 ZnC , j˛j D niD1 ˛ i , ˛Š D ˛ 1 Š : : : ˛ n Š, .x x0 /˛ D .x1 x01 /˛1 : : : qP n 2 .xn x0n /˛ n ). Here jj stands for the Euclidean norm: jxj D i D1 xi , x WD .x1 ; : : : ; xn / : Next we mention the Sobolev representations. Theorem 2.6. Let ˝ Rn be a domain star-shaped with respect to the open n ball R B D B .x0 ; r/ such that B l˝, ! 2 L1 .R /, the support supp ! B, ! .x/ dx D 1, l 2 N and f 2 C .˝/. Then for every x2˝ Rn f .x/ D
X 1 Z X 1 Z .D ˛ f / .y/ .x y/˛ ! .y/ dy C l .x y/˛ ! .y/ ˛Š B ˛Š B
j˛j
j˛jDl
Z
1
.1 t/l1 .D ˛ f / .y C t .x y// dt dy:
(2.4)
0
Proof. We write (2.3) for x; x0 D y, multiply it both sides by ! .y/ and integrate on B with respect to y. ˚ ˛ Call kD ˛ f kmax 1;l;B WD max kD f k1;B , where kk1;B is the supremum norm j˛jDl
on B, d WD diameter of B:
38
2 Multivariate Inequalities Based on Sobolev Representations
Proposition 2.7. Same assumption as in Theorem 2.6, x 2 B. Then R1 WD jRemainder (4)j
.nd /l k!kL1 .Rn / kD ˛ f kmax 1;l;B lŠ
:
(2.5)
Proof. We have that ˇ ˇ ˇ Z 1 ˇ Z ˇ ˇ X 1 ˇl .x y/˛ ! .y/ .1 t/l1 .D ˛ f / .y C t .x y// dt dy ˇˇ ˇ 0 ˇ ˇ j˛jDl ˛Š B l
Z 1 X 1 Z .1t/l1 j.D ˛ f / .yCt .xy//j dt dy j.xy/˛ j j! .y/j ˛Š B 0
j˛jDl
0 kD ˛ f
kmax 1;l;B
1 X 1 Z ˛ @ j.x y/ j j! .y/j dy A ˛Š B j˛jDl
1 X 1 Z dl @ j! .y/j dy A ˛Š B 0
kD ˛ f kmax 1;l;B
j˛jDl
0 1 max l X f d k!kL1 .B/ kD k1;l;B lŠ A @ D lŠ ˛Š ˛
j˛jDl
D
kD ˛ f
kmax 1;l;B
l
.d n/ k!kL1 .B/ : lŠ
From [6], p. 104, we mention Theorem 2.8. Let ˝ Rn be a domain star-shaped with respect to the open ball B D B .x0 ; r/ such that B ˝, Z ! 2 L1 .Rn / ; supp ! B, ! .x/ dx D 1; (2.6) Rn
l 2 N and f 2 C l .˝/. Then for every x 2 ˝ X 1 Z .D ˛ f / .y/ .x y/˛ ! .y/ dy f .x/ D ˛Š B j˛j
C
XZ j˛jDl
.D ˛ f / .y/ Vx
jx yjnl
w˛ .x; y/ dy;
(2.7)
2.2
Background
39
where for x; y 2 Rn , x ¤ y; j˛j .x y/˛ w .x; y/ ; ˛Š jx yjj˛j
(2.8)
yx n1 d ; ! xC jy xj jxyj
(2.9)
w˛ .x; y/ WD and
Z w .x; y/ WD
1
(for x D y 2 ˝ we define w˛ .x; x/ D w .x; x/ D 0). Remark 2.9. By (2.4) and (2.7) we derive Z 1 X 1 Z ˛ l1 ˛ .x y/ ! .y/ .1 t/ .D f / .y C t .x y// dt dy l ˛Š B 0 j˛jDl
D
XZ j˛jDl
.D ˛ f / .y/ Vx
jx yjnl
w˛ .x; y/ dy:
(2.10)
By Proposition 2.7 we obtain ˇ ˇ ˇ ˇX Z ˛ ˇ .nd /l k!kL1 .Rn / kD ˛ f kmax ˇ .D f / .y/ 1;l ˇ ˇ w .x; y/ dy : ˛ ˇ ˇ nl lŠ ˇ ˇj˛jDl Vx jx yj
(2.11)
Remark 2.10. Let D D diameter of ˝ be finite, i.e., ˝ is bounded. By [6] we have kw .x; y/kC .Rn Rn / k!kL1 .Rn / D n1 d;
(2.12)
and 8 ˛ 2 ZnC satisfying j˛j D l kw˛ .x; y/kC .Rn Rn / k!kL1 .Rn / nD n1 d:
(2.13)
Notice kw .x; y/kC .Rn Rn / k!kL1 .Rn / d n and kw˛ .x; y/kC .Rn Rn / k!kL1 .Rn / nd n , if ˝ D B: Hence, if ! is bounded, then for bounded ˝ the functions w and w˛ are bounded on Rn Rn . Also by [6], if ˝ is unbounded, then w, w˛ are bounded on K Rn for each compact K: If ! 2 C 1 .Rn /, then w .x; y/, w˛ .x; y/ have continuous derivatives of all orders 8 x; y 2 Rn W x ¤ y and at the points .x; x/, where x … B they are discontinuous, see [6].
40
2 Multivariate Inequalities Based on Sobolev Representations
Finally, we give the very general Sobolev representation, see [6]. Theorem 2.11. Let ˝ Rn be a domain star-shaped with respect to the open ball B D B .x0 ; r/ such that B ˝, Z n ! 2 L1 .R / ; supp ! B; ! .x/ dx D 1; (2.14) Rn
loc .˝/. Then for almost every x 2 ˝ l 2 N and f 2 W1l X 1 Z f .x/ D Dw˛ f .y/ .x y/˛ ! .y/ dy ˛Š B j˛j
˛ Dw f .y/
XZ
C
Vx
j˛jDl
jx yjnl
w˛ .x; y/ dy:
(2.15)
Corollary 2.12 ([6]). Let ˝ Rn be a domain star-shaped with respect to the open ball B D B .x0 ; r/ such that B ˝, Z ! 2 C01 .˝/ ; supp ! B; ! .x/ dx D 1: (2.16) Rn
loc Then 8 f 2 C l .˝/ for every x 2 ˝ and 8 f 2 W1l .˝/ for almost every x2˝ 1 0 Z X .1/j˛j @ Dy˛ Œ.x y/˛ ! .y/A f .y/ dy f .x/ D ˛Š B j˛j
C
XZ j˛jDl
.D ˛ f / .y/ jx yjnl
Vx
w˛ .x; y/ dy
(2.17)
loc with Dw˛ f replacing D ˛ f in the case of f 2 W1l .˝/ : Next ˛ ˇ means ˛ i ˇ i ; i D 1; :::; n and ˛ ˇ 2 ZC : Corollary 2.13 ([6]). Under the assumptions of Corollary 2.12, let ˇ 2 ZnC and loc .˝/ for 0 < jˇj < l. Then 8 f 2 C l .˝/ for every x 2 ˝ and 8 f 2 W1l almost every x 2 ˝ 1 0 Z X .1/j˛jCjˇj ˇ @ D f .x/ D Dy˛Cˇ Œ.x y/˛ ! .y/A f .y/ dy ˛Š B j˛j
C
X j˛jDl;˛ˇ
Z
.D ˛ f / .y/ Vx
jx yjnlCjˇj
w˛ˇ .x; y/ dy
loc ˇ .˝/ : with Dw f replacing D ˇ f and Dw˛ f replacing D ˛ f if f 2 W1l
(2.18)
2.2
Background
41
Remark 2.14. Again d D diamB, D D diam˝. We suppose k!kL1 .Rn / < 1. ˛ Here D f could mean either D ˛ f or Dw˛ f . Then ˛ ˇ ˇ ˇX Z ˇ D f .y/ ˇ ˇ ˇ R2 WD ˇ w˛ .x; y/ dy ˇˇ nl ˇj˛jDl Vx jx yj ˇ
XZ j˛jDl
0
ˇ ˇ ˇ ˇ ˛ jx yjln ˇ D f .y/ˇ jwa .x; y/j dy
Vx
XZ
@
j˛jDl
Vx
1 ˇ ˇ ˛ ˇ ˇ jx yjln ˇ D f .y/ˇ dy A k!kL1 .Rn / ndD n1 ;
(2.19)
if D < 1: Next we assume x 2 B, l n, then we retake ˝ D B, i.e., d D D, etc., and thus ˇ X Z ˇˇ ˛ ˇ R2 ˇ D f .y/ˇ dy k!kL1 .Rn / nd l j˛jDl
B
0
X ˛ D@ D f j˛jDl
1 L1 .B/
A k!kL
1 .R
n/
nd l :
That is, we proved for l n, x 2 B, that ˇ ˇ 0 1 ˇ ˇ Z D ˛ f .y/ X ˇX ˇ ˛ ˇ A k!kL .Rn / nd l : w˛ .x; y/ dy ˇˇ @ D f ˇ 1 nl .B/ L 1 yj jx B ˇj˛jDl ˇ j˛jDl (2.20) ˛ < 1 for all ˛ W j˛j D l (which Again we assume x 2 B, l n and D f L1 .B/
is true for D ˛ f always by f 2 C l .˝/). Then 0
X ˛ R2 @ D f j˛jDl
1
L1 .B/
We know that
A Vol .B/ k!kL
n
Vol .B/ D
1 .R
n
n/
nd l DW ./ :
dn 2 2 n rn D n n; 2 C1 2 C1 2
where is the gamma function.
42
2 Multivariate Inequalities Based on Sobolev Representations
Therefore 0
X ˛ ./ D @ D f j˛jDl
L1 .B/
1 A k!kL
n
1
.Rn /
˛ So we have proved if l n, x 2 B, and D f
n 2 d lCn : n 2 n2 C 1
L1 .B/
˛ ˇ ˇ ˇ ˇ Z D f .y/ ˇX ˇ ˇ w˛ .x; y/ dy ˇˇ ˇ nl ˇj˛jDl B jx yj ˇ 1 0 X ˛ A k!kL .Rn / @ D f 1 j˛jDl
L1 .B/
< 1 for all ˛ W j˛j D l that
n
n 2 d lCn : n 2 n2 C 1
(2.21)
We use Lemma (4.3.1), p. 100 of [5]. It follows Lemma 2.15. If f 2 Lp .˝ /, 1 < p < 1, ˝ is a region of diameter d > 0, and m > pn , then Z
m pn
˝
jx zjmn jf .z/j dz cp d
kf kLp .˝ / ;
(2.22)
8 x 2 ˝ , where cp is a constant depending only on p: We make ˛
Remark 2.16 (continuing from Remark 2.14). We assume now that D f 2 Lp .B/, j˛j D l, 1 < p < 1, l > pn . Then by (2.22), Z B
ˇ ˇ ˇ ˛ ˇ l n ˛ jx yjln ˇ D f .y/ˇ dy cp d p D f
; x 2 B:
(2.23)
ˇ ˇ ˇ ˇ Z D ˛ f .y/ ˇ ˇX ˇ ˇ w .x; y/ dy ˛ ˇ ˇ nl ˇ ˇj˛jDl B jx yj 1 0 X n ˛ A k!kL .Rn / ncp d l p Cn ; x 2 B: @ D f 1
(2.24)
Lp .B/
Consequently, we derive
j˛jDl
Lp .B/
2.2
Background
43
Also we make Remark 2.17. Again here d D diamB, assume ! 2 C01 .˝/, supp ! B, i.e., R ˛ k!k1 < 1, Rn ! .x/ dx D 1. Here D f is either D ˛ f or Dw˛ f . Then for l n C jˇj ; x 2 B, we obtain ˛ ˇ ˇ ˇ ˇ Z D f .y/ ˇ X ˇ ˇ w .x; y/ dy ˇˇ ˇ nlCjˇj ˛ˇ ˇj˛jDl;˛ˇ B jx yj ˇ 1 0 X ˛ A k!k1 nd ljˇj : @ D f ˛ Also for l n C jˇj, D f
(2.25)
L1 .B/
j˛jDl;˛ˇ
L1 .B/
< 1, j˛j D l, ˛ ˇ, x 2 B, we get
˛ ˇ ˇ ˇ X Z ˇ D f .y/ ˇ ˇ ˇ ˇ w .x; y/ dy ˇ ˇ nlCjˇj ˛ˇ ˇj˛jDl;˛ˇ B jx yj ˇ 1 0 n X n 2 ˛ A k!k1 n d ljˇjCn : @ D f L1 .B/ 2n 2 C 1
(2.26)
j˛jDl;˛ˇ
˛
Next, suppose D f 2 Lp .B/, all ˛ W j˛j D l, ˛ ˇ, 1 < p < 1, l > x 2 B: Then ˛ ˇ ˇ ˇ ˇ Z D f .y/ ˇ X ˇ ˇ ˇ w .x; y/ dy ˇ ˇ nlCjˇj ˛ˇ ˇj˛jDl;˛ˇ B jx yj ˇ 1 0 X n ˛ A k!k1 ncp d ljˇj p Cn : @ D f j˛jDl;˛ˇ
Lp .B/
n p
C jˇj,
(2.27)
We make ˛
Remark 2.18. Here D f denotes either D ˛ f or Dw˛ f , d D diamB: Suppose k!kL1 .Rn / < 1: Denote by Ql1 f .x/ WD
X 1j˛jl1
1 ˛Š
Z ˛ D f .y/ .x y/˛ ! .y/ dy; 8 x 2 ˝; (2.28) B
the quasi-averaged Taylor polynomial. When l D 1, then Q 0 f .x/ WD 0: X In this chapter sums of the form : D 0: 1j˛j0
44
2 Multivariate Inequalities Based on Sobolev Representations
Then for x 2 B we obtain 1 0 ˇ X 1 Z ˇˇ ˛ ˇ ˇ l1 ˇ ˇQ f .x/ˇ @ ˇ D f .y/ˇ j.x y/˛ j dy A k!kL1 .Rn / ˛Š B 0
1j˛jl1
X
@
1j˛jl1
1 ˛Š
1 Z ˇ ˇ ˇ ˛ ˇ ˇ D f .y/ˇ jx yjj˛j dy A k!kL1 .Rn / B
DW ./ : We notice that
0
(2.29)
X
./ @
d j˛j ˛Š
1j˛jl1
D
8 <
X
j˛j
d ˛Š
:
1j˛jl1
!1 Z ˇ ˇ ˇ ˇ ˛ ˇ D f .y/ˇ dy A k!kL1 .Rn / B
˛ D f
L1 .B/
!9 = ;
k!kL1 .Rn / :
So we have proved for x 2 B that 8 !9 = j˛j ˇ l1 ˇ < X d ˛ ˇQ f .x/ˇ k!kL1 .Rn / : D f : L1 .B/ ; ˛Š
(2.30)
1j˛jl1
˛ Also, when D f
L1 .B/
0 @
X
j˛j
d ˛Š
1j˛jl1
0
!1 Z ˇ ˇ ˇ ˛ ˇ ˇ D f .y/ˇ dy A k!kL1 .Rn /
X
@
1j˛jl1
0 D@
< 1, for all ˛ W 1 j˛j l 1, we get
X
1j˛jl1
B
!1 d j˛j ˛ A Vol .B/ k!kL .Rn / D f 1 L1 .B/ ˛Š !1 n dn d j˛j 2 ˛ A n: k!kL1 .Rn / n D f L1 .B/ ˛Š 2 C1 2
˛ So we proved, when D f
L1 .B/
0 ˇ l1 ˇ ˇQ f .x/ˇ @
X
1j˛jl1
< 1, for all ˛ W 1 j˛j l 1, x 2 B, that
!1 n k!kL1 .Rn / 2 d .nCj˛j/ ˛ A n : (2.31) D f L1 .B/ ˛Š 2n 2 C 1
2.2
Background
45
We need Lemma 2.19. Let ˝ be a region of Rn of finite diameter d > 0 and f 2 Lp .˝ /, 1 < p; q < 1 W p1 C q1 D 1 and m 2 N, then Z ˝
m
jx zj jf .z/j dz cq;m;n d
mC nq
kf kLp .˝ / ; 8 x 2 ˝ :
(2.32)
Proof. We see that Z ˝
jx zjm jf .z/j dz
Z ˝
q1 jx zjmq dz kf kLp .˝ /
(using polar coordinates) Z Cn
d
! q1 r mqCn1 dr
kf kLp .˝ /
0 mqCn q
D cq;m;n d
kf kLp .˝ / D
mC nq cq;m;n d
kf kLp .˝ / : 1 p
1 q
Remark 2.20 (continuing from Remark 2.18). Let p; q > 1 W C D 1. Assume ˛ D f 2 Lp .B/, for all 1 j˛j l 1, x 2 B, then by Lemma 2.19 we obtain 0 ˇ ˇ l1 ˇQ f .x/ˇ @ 0 @
X
1j˛jl1
X 1j˛jl1
1 ˛Š
1 Z ˇ ˇ ˇ ˛ ˇ ˇ D f .y/ˇ jx yjj˛j dy A k!kL1 .Rn / B
1 cq;j˛j;n d ˛Š
j˛jC nq
˛ D f
Lp .B/
1 A k!kL
1 .R
n/
:
That is 0 B ˇ ˇ l1 ˇQ f .x/ˇ cq;l;n k!k .Rn / B L1 @
X
1j˛jl1
x 2 B:
11 0 ˛ j˛jC nq f d D B CC Lp .B/ B CC ; @ AA ˛Š
(2.33)
Remark 2.21. For x 2 B, we consider here ! 2 C01 .˝/, supp ! B, R l loc ˛ l .˝/. Here D denotes any of Rn ! .x/ dx D 1; f 2 C .˝/ or f 2 W1 D ˛ , Dw˛ : We also consider
46
2 Multivariate Inequalities Based on Sobolev Representations
Ql1 f .x/ D
0
Z
@ B
D
X
1j˛jl1
X 1j˛jl1
.1/j˛j ˛Š
Hence ˇ ˇ l1 ˇQ f .x/ˇ
X
1 ˛Š
1j˛jl1
1 .1/j˛j ˛ Dy Œ.x y/˛ ! .y/A f .y/ dy ˛Š Z B
Dy˛ Œ.x y/˛ ! .y/ f .y/ dy:
Z ˇ ˇ ˇ ˛ ˇ ˇDy Œ.x y/˛ ! .y/ˇ jf .y/j dy:
Let ˇ 2 ZnC and 0 < jˇj < l. We consider here 0 Z X @ Qˇl1 f .x/ WD 1j˛jljˇj1
X
D
1j˛jljˇj1
(2.35)
B
We derive 8 x 2 B, 1 80 X ˆ ˆ 1 ˛ ˆ @ ˆ Œ.x y/˛ ! .y/ A kf kL1 .B/ ; ˆ ˛Š Dy ˆ 1 ˆ ˆ ˆ 1 01j˛jl1 ˆ ˆ ˆ X ˆ ˆ 1 ˛ ˆ A kf kL .B/ ; @ ˆ Œ.x y/˛ ! .y/ ˆ ˛Š Dy 1 ˆ L1 .B/ < ˇ l1 ˇ 1j˛jl1 ˇQ f .x/ˇ if f 2 L1 .B/ ; ˆ ˆ ˆ 1 1 ˆ when p; q > 1 W p C q D 1, we have ˆ ˆ 1 0 ˆ ˆ ˆ ˆ X ˆ 1 ˛ ˆ A kf kL .B/ ; ˆ Œ.x y/˛ ! .y/ ˆ@ ˛Š Dy p ˆ Lq .B/ ˆ ˆ 1j˛jl1 ˆ : if f 2 Lp .B/ :
B
(2.34)
(2.36)
1 .1/j˛jCjˇj ˛Cˇ Dy Œ.x y/˛ ! .y/A f .y/ dy ˛Š
.1/j˛jCjˇj ˛Š
Z B
Dy˛Cˇ Œ.x y/˛ ! .y/
f .y/ dy:
(2.37)
When l D jˇj C 1, then Qˇl1 f .x/ WD 0. Hence ˇ ˇ ˇ ˇ l1 ˇQˇ f .x/ˇ
X 1j˛jljˇj1
1 ˛Š
Z ˇ ˇ ˇ ˛Cˇ ˇ ˇDy Œ.x y/˛ ! .y/ˇ jf .y/j dy: (2.38) B
2.2
Background
47
We obtain 8 x 2 B; 1 80 ˆ X ˆ ˛Cˇ ˆ ˛ 1 @ ˆ Œ.x y/ ! .y/ A kf kL1 .B/ ; ˆ ˛Š Dy ˆ 1 ˆ ˆ 1j˛jljˇj1 ˆ ˆ ˆ ˆ ˆ 0 1 ˆ ˆ ˆ ˆ X ˆ ˛Cˇ ˆ ˛ 1 ˆ @ A kf kL .B/ ; Œ.x y/ ! .y/ ˆ ˛Š Dy 1 ˆ L1 .B/ < 1j˛jljˇj1 ˇ ˆ ˇ ˇ ˇ l1 ˇQˇ f .x/ˇ if f 2 L1 .B/ ; ˆ ˆ ˆ ˆ 1 1 ˆ ˆ when p; q > 1 W p C q D 1, we have ˆ ˆ ˆ ˆ ˆ 1 0 ˆ ˆ ˆ X ˆ ˆ 1 ˛Cˇ ˆ A kf kL .B/ ; @ Œ.x y/˛ ! .y/ ˆ ˛Š Dy p ˆ ˆ Lq .B/ ˆ 1j˛jljˇj1 ˆ ˆ : if f 2 Lp .B/ : (2.39) The final remark follows. ˛
ˇ
ˇ
Remark 2.22. Here D denotes D ˛ or Dw˛ , and D means D ˇ or Dw : We rewrite (2.4), (2.7), (2.15), and (2.17). For x 2 ˝ we get Z
f .y/ ! .y/ dy C Ql1 f .x/ C Rl f .x/ ;
f .x/ D
(2.40)
B
where Rl f .x/ WD
XZ
˛
D f .y/ Vx
j˛jDl
jx yjnl
w˛ .x; y/ dy
(2.41)
is equal to the remainders of ((2.4)and (2.44), respectively). Also for x 2 ˝ we write (2.18) as follows: Z ˇ ˇ D f .x/ D .1/jˇj D ! .y/ f .y/ dy C Qˇl1 f .x/ C Rˇl f .x/ ; (2.42) B
where Rˇl f .x/ WD
X j˛jDl;˛ˇ
Z Vx
˛ D f .y/
jx yjnlCjˇj
w˛ˇ .x; y/ dy:
(2.43)
48
2 Multivariate Inequalities Based on Sobolev Representations
Additionally we give Theorem 2.23. Let ˝ Rn be a domain star-shaped with respect R to the open ball B D B .x0 ; r/ such that B ˝, ! 2 L1 .Rn /, supp ! B, Rn ! .x/ dx D 1, loc .˝/. Then for almost every x 2 ˝ l 2 N and f 2 W1l X 1 Z f .x/ D Dw˛ f .y/ .x y/˛ ! .y/ dy ˛Š B j˛j
Cl
X 1 Z .x y/˛ ! .y/ ˛Š B
j˛jDl
Z
1
.1 t/
l1
0
Dw˛ f
.y C t .x y// dt dy:
(2.44)
Proof. From the assumptions of the theorem, we get for almost every x 2 ˝ that f .x/
X 1 Z Dw˛ f .y/ .x y/˛ ! .y/ dy ˛Š B
j˛j
D
XZ j˛jDl
Vx
Dw˛ f .y/
jx yjnl
w˛ .x; y/ dy;
R .D ˛ f /.y/ implying that Vx w nl w˛ .x; y/ dy is finite for almost every x 2 ˝. jxyj From [6], p. 105, (3.41) there, we know that 8 x 2 Rn sup py w˛ .x; y/ D sup py w .x; y/ Kx ; where Kx is the cone in Rn related to Vx , see again [6], pp. 93–100. So acting similarly to [6], p. 107 and working backwards we derive
XZ j˛jDl
Vx
Dw˛ f .y/
jx yj
nl
w˛ .x; y/ dy D l
X 1 J˛ ; ˛Š
j˛jDl
where Z J˛ D
˛ .x z/˛ Dw f .z/ jx zjn Rn
Replacing by
jxzj , 1t
Z J˛ D
Rn
zx n1 d dz: ! xC jz xj jxzj
Z
1
we obtain
˛ Dw f .z/ .x z/˛
Z
1
! 0
z tx 1t
dt .1 t/nC1
dz:
2.2
Background
49
Next, setting z D y C t .x y/ and noticing .x z/˛ D .1 t/l .x y/˛ and dz D .1 t/n dy we find that Z J˛ D
1
.1 t/l1
Z Rn
0
Z D
Rn
Z
.x y/˛ ! .y/
.x y/˛ ! .y/
D
Dw˛ f .y C t .x y// .x y/˛ ! .y/ dy dt
Z
1 0
Z
1 0
B
.1 t/l1 Dw˛ f .y C t .x y// dt dy
.1 t/l1 Dw˛ f .y C t .x y// dt dy:
We have proved that
XZ j˛jDl
Dw˛ f .y/
jx yjnl
Vx
w˛ .x; y/ dy
Z 1 X 1 Z ˛ ˛ l1 Dw f .y C t .x y// dt dy; Dl .x y/ ! .y/ .1 t/ ˛Š B 0 j˛jDl
establishing the claim.
Proposition 2.24. Same assumptions as in Theorem 2.23. Then for almost every x 2 B we get jRemainder (2.44)j ld l k!kL1 .Rn /
X 1 D ˛ f .y C t .x y// : w L1 .BŒ0;1/ ˛Š
j˛jDl
(2.45) In (2.45) we assume for all ˛ W j˛j D l that ˛ D f .y C t .x y// w L
1 .BŒ0;1/
< 1:
Proof. We have that ˇ ˇ ˇ X Z 1 ˇ Z ˇ ˇ 1 ˇl .x y/˛ ! .y/ .1 t/l1 Dw˛ f .y C t .x y// dt dy ˇˇ ˇ 0 ˇ ˇ j˛jDl ˛Š B Z X 1 l j.x y/˛ j j! .y/j ˛Š B j˛jDl
Z
0
1
.1 t/
l1
ˇ ˇ ˛ ˇ D f .y C t .x y//ˇ dt dy w
50
2 Multivariate Inequalities Based on Sobolev Representations
X 1 Z Z 1 ˇ ˇ ˛ l1 ˇ ˇ .1 t/ Dw f .y C t .x y// dt dy ˛Š B 0
l
ld k!kL1 .Rn /
j˛jDl
ld l k!kL1 .Rn /
X 1 Z Z 1 ˇ ˇ ˇ D ˛ f .y C t .x y//ˇ dt dy w ˛Š B 0
j˛jDl
X 1 D ˛ f .y C t .x y// ; w L1 .BŒ0;1/ ˛Š
D ld l k!kL1 .Rn /
j˛jDl
proving the claim.
2.3 Main Results On the way to prove the general Chebyshev–Gr¨uss-type inequalities, we establish the general Theorem 2.25. For f, g under the assumptions of any of Theorems 2.6, 2.8, 2.11, 2.23 and Corollary 2.12 we obtain that ˇZ Z Z ˇ ˇ ˇ ˇ ! .x/ f .x/ dx ! .x/ g .x/ dx ˇˇ .f; g/ WDˇ ! .x/ f .x/ g .x/ dx B
1 2
B
Z
B
ˇ ˇ j! .x/j jg .x/j ˇQ l1 f .x/ˇ dx
B
Z
ˇ ˇ j! .x/j jf .x/j ˇQl1 g .x/ˇ dx
C
B
Z
ˇ ˇ j! .x/j jg .x/j ˇRl f .x/ˇ dx
C Z
B
C
ˇ ˇ j! .x/j jf .x/j ˇRl g .x/ˇ dx
:
B
Proof. For x 2 B we have Z
f .y/ ! .y/ dy C Ql1 f .x/ C Rl f .x/ ;
f .x/ D B
and
Z g .x/ D B
g .y/ ! .y/ dy C Ql1 g .x/ C Rl g .x/ :
(2.46)
2.3
Main Results
51
Hence ! .x/ f .x/ g .x/ Z D ! .x/ g .x/ f .y/ ! .y/ dy C! .x/ g .x/ Ql1 f .x/C! .x/ g .x/ Rl f .x/; B
and ! .x/ f .x/ g .x/ D ! .x/ f .x/
Z
g .y/ ! .y/ dy C! .x/ f .x/ Ql1 g .x/C! .x/ f .x/ R l g .x/ :
B
Therefore Z Z Z ! .x/ f .x/ g .x/ dx D ! .x/ g .x/ dx f .y/ ! .y/ dy B
B
B
Z
! .x/ g .x/ Ql1 f .x/ dx
C B
Z
! .x/ g .x/ Rl f .x/ dx;
C B
and
Z
Z
Z
! .x/ f .x/ g .x/ dx D
! .x/ f .x/ dx
B
g .y/ ! .y/ dy
B
Z
! .x/ f .x/ Q l1 g .x/ dx
C Z
B
B
! .x/ f .x/ Rl g .x/ dx:
C B
Consequently, there hold Z
Z
Z
! .x/ f .x/ g .x/ dx B
Z
D
! .x/ g .x/ dx B
! .x/ g .x/ Ql1 f .x/ dx C
Z
B
f .x/ ! .x/ dx B
! .x/ g .x/ Rl f .x/ dx;
B
and Z
Z
Z
! .x/ f .x/ g .x/ dx B
Z
D B
! .x/ f .x/ dx B
! .x/ f .x/ Q l1 g .x/ dx C
g .x/ ! .x/ dx B
Z
! .x/ f .x/ Rl g .x/ dx: B
52
2 Multivariate Inequalities Based on Sobolev Representations
Adding the last two equalities we obtain Z Z Z ! .x/ f .x/ g .x/ dx ! .x/ f .x/ dx ! .x/ g .x/ dx B
1 D 2
B
Z
! .x/ g .x/ Ql1 f .x/ dx C
B
Z
B
! .x/ f .x/ Q l1 g .x/ dx
B
Z
Z
l
C
! .x/ g .x/ R f .x/ dx C B
l
! .x/ f .x/ R g .x/ dx
;
B
hence proving the claim. We give Theorem 2.26. Let ˝ Rn be a domain star-shaped with respect R to the open ball B D B .x0 ; r/ such that B ˝, ! 2 L1 .Rn /, supp ! B, Rn ! .x/ dx D 1; l 2 N and f; g 2 C l .˝/ : Then ˇZ Z Z ˇ ˇ ˇ ˇ ˇ ! .x/ f .x/ g .x/ dx ! .x/ f .x/ dx ! .x/ g .x/ dx ˇ ˇ B B B 22 1 0 X d j˛j kD ˛ f k1;B k!k2L1 .Rn / 44kgk1;B @ A 2 ˛Š 1j˛jl1
0
X
C kf k1;B @
1j˛jl1
13 d j˛j kD ˛ gk1;B A5 ˛Š
#3 .nd /l max ˛ 5; C kgk1;B kD ˛ f kmax 1;l;B C kf k1;B kD gk1;l;B lŠ "
(2.47) where d is the diameter of B. When l D 1 the sums in (2.47) collapse. Proof. One also in general obtains (x 2 B) ˇ l1 ˇ ˇQ f .x/ˇ
X 1j˛jl1
X 1j˛jl1
D
X 1j˛jl1
1 ˛Š 1 ˛Š
Z ˇ ˇ ˇ ˛ ˇ ˇD f .y/ˇ j.x y/˛ j j! .y/j dy B
Z B
˛ j! .y/j dy d j˛j D f
1 ˛ : k!kL1 .B/ d j˛j D f L1 .B/ ˛Š
L1 .B/
2.3
Main Results
53
˛ So for D f
L1 .B/
< 1, for all ˛ W 1 j˛j l 1, we proved ˛ d j˛j D f
0 X
ˇ l1 ˇ B ˇQ f .x/ˇ k!k L1 .B/ @
˛Š
1j˛jl1
1
L1 .B/ C
A;
(2.48)
for x 2 B. By (2.46), Proposition 2.7 and (2.48) we obtain 2 0 2 1 .f; g/ 4k!k2L1 .B/ 4kgk1;B @ 2
1 d j˛j kD ˛ f k1;B A ˛Š
X
1j˛jl1
0
X
C kf k1;B @
1j˛jl1
" C k!kL1 .B/ kgk1;B
C kf k1;B
D
k!k2L1 .Rn / 2
22
13 d j˛j kD ˛ gk1;B A5 ˛Š
.nd /l k!kL1 .Rn / kD ˛ f kmax 1;l;B lŠ
.nd /l k!kL1 .Rn / kD ˛ gkmax 1;l;B lŠ 0
44kgk1;B @
X 1j˛jl1
0 C kf k1;B @
#3 5
1 d j˛j kD ˛ f k1;B A ˛Š
X 1j˛jl1
13 d j˛j kD ˛ gk1;B A5 ˛Š
#3 .nd /l max ˛ 5; C kgk1;B kD ˛ f kmax 1;l;B C kf k1;B kD gk1;l;B lŠ "
proving the claim. We present Theorem 2.27. Let ˝ Rn be a domain star-shaped with respect R to the open ball B D B .x0 ; r/ such that B ˝, ! 2 L1 .Rn /, supp ! B, Rn ! .x/ dx D 1; loc .˝/ : Suppose further that l n. Then l 2 N and f; g 2 W1l
54
2 Multivariate Inequalities Based on Sobolev Representations
ˇZ Z Z ˇ ˇ ˇ ˇ ! .x/ f .x/ g .x/ dx ! .x/ f .x/ dx ! .x/ g .x/ dx ˇˇ ˇ B
B
k!k2L1 .Rn / 2
B
0 28 < 4 kgkL .B/ @ 1 :
!1 d j˛j D ˛ f A w L1 .B/ ˛Š
X
1j˛jl1
0
! 19 = d j˛j D ˛ g A w L1 .B/ ; ˛Š
X
C kf kL1 .B/ @
1j˛jl1
2
0
1 X D ˛ f A C .nd / 4kgkL1 .B/ @ w L1 .B/ l
j˛jDl
0
133 X D ˛ g A55 : (2.49) C kf kL1 .B/ @ w L1 .B/ j˛jDl
Proof. Here we get by (2.43), (2.30), and (2.20) that
.f; g/
k!k2L1 .Rn / 2
0 28 < 4 kgkL .B/ @ 1 :
X
1j˛jl1
0
!1 d j˛j D ˛ f A w L1 .B/ ˛Š
X
C kf kL1 .B/ @
1j˛jl1
!19 = d j˛j D ˛ g A w L1 .B/ ; ˛Š
82 0 1 < X D ˛ f A C 4kgkL1 .B/ @ w L1 .B/ : j˛jDl
93 13 = X l 5 D ˛ g A 5 C kf kL1 .B/ @ ; nd w L1 .B/ ; 0
j˛jDl
proving the claim. Based on (2.46), (2.31), and (2.21) we have Theorem 2.28. Let ˝ Rn be a domain star-shaped with respect to the open ball B D B .x0 ; r/ such that B ˝, ! 2 L1 .Rn /, supp ! B,
2.3
Main Results
55
loc R l 2 N and f; g 2 W1l .˝/ : Furthermore assume dx D 1; Rn ˛! .x/ D ˛ g D f , < 1 for all ˛ W 1 l; l n. Then j˛j w w L1 .B/ L1 .B/ ˇZ Z Z ˇ ˇ ˇ ˇ ! .x/ f .x/ g .x/ dx ! .x/ f .x/ dx ! .x/ g .x/ dx ˇˇ ˇ B B B 82 0 !1 n nCj˛j X 2 k!k2L1 .Rn / < d D ˛ f A 4kgkL1 .B/ @ nC1 n w L1 .B/ ˛Š 2 2 C1 : 1j˛jl1
0
!13 d nCj˛j D ˛ g A5 w L1 .B/ ˛Š
X
C kf kL1 .B/ @
1j˛jl1
0
1 X D ˛ f A C 4kgkL1 .B/ @ w L1 .B/ 2
j˛jDl
9 13 = X lCn D ˛ g A 5 nd C kf kL1 .B/ @ : (2.50) w L1 .B/ ; 0
j˛jDl
Based on (2.46), (2.33), and (2.24) we get Theorem 2.29. Let ˝ Rn be a domain star-shaped with respect R to the open ball B D B .x0 ; r/ such that B ˝, ! 2 L1 .Rn /, supp ! B, Rn ! .x/ dx D 1; loc .˝/ : Furthermore suppose p; q > 1 W p1 C q1 D 1; l > pn ; l 2 N and f; g 2 W1l for all ˛ W 1 j˛j l; Dw˛ f , Dw˛ g 2 Lp .B/. Then ˇZ Z Z ˇ ˇ ˇ ˇ ! .x/ f .x/ g .x/ dx ˇ ! .x/ f .x/ dx ! .x/ g .x/ dx ˇ ˇ B B B 82 0 1 X k!k2L1 .Rn / < D ˛ f A 4kgkL .B/ @ w 1 Lp .B/ : 2 j˛jDl
0
X D ˛ g C kf kL1 .B/ @ w L 2
j˛jDl
0
C cq;l;n 4kgkL1 .B/ @
13 n
p .B/
X 1j˛jl1
0 C kf kL1 .B/ @
A5 ncp d l p Cn
11 0 j˛jC nq D ˛ f d w Lp .B/ AA @ ˛Š
X 1j˛jl1
0 1139 j˛jC nq D ˛ g = d w Lp .B/ @ AA5 : ; ˛Š (2.51)
56
2 Multivariate Inequalities Based on Sobolev Representations
Remark 2.30. When f; g 2 C l .˝/ the Theorems 2.27, 2.28, 2.29 are again valid. In this case we replace Dw˛ by Dw in all inequalities (2.49), (2.50), and (2.51). We give to the open ball Theorem 2.31. Let ˝ Rn be a domain star-shaped with respect R B D B .x0 ; r/ such that B ˝, ! 2 C01 .˝/, supp ! B, Rn ! .x/ dx D 1: Let loc ˛ f; g either in C l .˝/ or in W1l .˝/. Here D f denotes either D ˛ f or Dw˛ f and .f; g/ is as in (2.46). We have the following cases: (i) Here l n. Then .f; g/
k!k1 2
28 < 4 2 kgkL .B/ kf kL .B/ 1 1 : 0 @
X
1j˛jl1
19 = 1 ˛ ˛ A Dy Œ.x y/ w .y/ 1;B 2 ; ˛Š 2
0
X ˛ C k!k1 nd l 4kgkL1 .B/ @ D f j˛jDl
0
X ˛ C kf kL1 .B/ @ D g j˛jDl ˛
L1 .B/
L1 .B/
1 A
133 A55 :
(2.52)
˛
(ii) Here l nI f; g 2 L1 .B/ and D f , D g 2 L1 .B/ for all ˛ W j˛j D l: Then .f; g/
k!k1 ˚ kf kL1 .B/ kgkL1 .B/ C kf kL1 .B/ kgkL1 .B/ 2 1 0 X 1 A sup Dy˛ Œ.x y/˛ w .y/ @ L1 .B/ ˛Š x2B 1j˛jl1
0 1 2 n X nd lCn 2 k!k1 4 ˛ A C n n kgkL1 .B/ @ D f L1 .B/ 2 2 C1 j˛jDl 0 139 = X ˛ A5 : C kf kL1 .B/ @ (2.53) D g ; L1 .B/ j˛jDl
2.3
Main Results
(iii) Let p; q > 1 W Lp .B/ : Then
57 1 p
C
1 q
n ; p
D 1; l >
˛
˛
for all ˛ W j˛j D l; D f , D g; f; g 2
2 k!k1 4 .f; g/ kgkL1 .B/ kf kLp .B/ C kf kL1 .B/ kgkLp .B/ 2 0
X
@
1j˛jl1
1 1 ˛ ˛ A sup Dy Œ.x y/ w .y/ Lq .B/ ˛Š x2B 2
0
X
C k!k1 cq;l;n 4kgkL1 .B/ @
1j˛jl1
0 C kf kL1 .B/ @
X 1j˛jl1
0 ˛ 11 D f Lp .B/ j˛jC n q @ AA d ˛Š
0 ˛ 1133 D g Lp .B/ j˛jC n q AA55 : @ d ˛Š
(2.54)
Proof. By use of (2.36) and Theorems 2.27–2.29. We also present Theorem 2.32. Let ˝ Rn be a domain star-shaped with respect R to the open ball B D B .x0 ; r/ such that B ˝, ! 2 L1 .Rn /, supp ! B, Rn ! .x/ dx D 1; loc .˝/ for all ˛ W j˛j D l l 2 Nand f;g 2 W1l : Furthermore suppose n that Dw˛ f .y C t .x y//L1 .BŒ0;1/ ; Dw˛ g .y C t .x y//L1 .BŒ0;1/ < 1. Then 0 28 !1 2 < j˛j X k!kL1 .Rn / d ˛ D f A .f; g/ 4 kgkL1 .B/ @ w L1 .B/ : 2 ˛Š 1j˛jl1
0
X
C kf kL1 .B/ @ 2
1j˛jl1
!19 = d j˛j D ˛ g A w L1 .B/ ; ˛Š
0
1 X 1 D ˛ f .y C t .x y// A Cld l 4kgkL1 .B/ @ w L1 .BŒ0;1/ ˛Š 0
j˛jDl
13 3 X 1 D ˛ g .y C t .x y// A5 5 C kf kL1 .B/ @ w L1 .BŒ0;1/ ˛Š j˛jDl
: Proof. By Theorem 2.27 and Proposition 2.24.
(2.55)
58
2 Multivariate Inequalities Based on Sobolev Representations
Next, we give a series of Ostrowski-type inequalities. Theorem 2.33. Let all as in Theorem 2.6. Call again Z X 1 l1 Q f .x/ WD .D ˛ f / .y/ .x y/˛ ! .y/ dy: ˛Š B 1j˛jl1
Then for every x 2 B, we obtain ˇ ˇ Z ˇ ˇ l1 ˇf .x/ f .y/ ! .y/ dy Q f .x/ˇˇ ˇ B
l
.nd / k!kL1 .Rn / kD ˛ f kmax 1;l;B WD A1 : lŠ
(2.56)
Also it holds, by additionally assuming k!kL1 .Rn / < 1, that ˇ ˇ Z ˇ ˇ ˇ ˇf .x/ f .y/ ! .y/ dy ˇ ˇ B 0 !1 n nCj˛j X k!kL1 .Rn / 2 d ˛ A @ kD f k1 ˛Š 2n n2 C 1 1j˛jl1
C
.nd /l k!kL1 .Rn / kD ˛ f kmax 1;l;B WD B1 : lŠ
(2.57)
Proof. Use of Theorem 2.6, (2.40), (2.5), and (2.31). We continue with Theorem 2.34. All as in Theorems 2.8 or 2.11. Assume l n, k!kL1 .Rn / < 1. Then for every x 2 B (almost every x 2 B, respectively) we get ˇ ˇ Z ˇ ˇ l1 ˇ E .f / .x/ WD ˇf .x/ f .y/ ! .y/ dy Q f .x/ˇˇ B
0
X ˛ @ D f j˛jDl
1 L1 .B/
A k!kL
1 .R
n/
nd l DW A2 :
(2.58)
Also it holds
ˇ ˇ Z ˇ ˇ .f / .x/ WD ˇˇf .x/ f .y/ ! .y/ dy ˇˇ B 28 !9 = < X j˛j d ˛ 4 D f : L1 .B/ ; ˛Š 1j˛jl1
0
X ˛ C@ D f j˛jDl
L1 .B/
1
3
A nd l 5 k!kL
1 .R
n/
DW B2 : (2.59)
2.3
Main Results
59
Proof. Use of Theorems 2.8, 2.11; (2.40), (2.20), (2.28), and (2.30). We give Theorem 2.35. All as in Theorems 2.8, 2.11. Suppose l n, k!kL1 .Rn / < 1; ˛ and D f < 1 for all ˛ W j˛j D l. Then for every x 2 B (almost every L1 .B/
x 2 B, respectively), it holds 0 X ˛ E .f / .x/ @ D f
1
L1 .B/
j˛jDl
1 .R
˛ Additionally, assume that D f 20 .f / .x/ 4@
L1 .B/
X
1j˛jl1
n
A k!kL
n/
n 2 d lCn DW A3 : (2.60) n 2 n2 C 1
< 1 for all ˛ W 1 j˛j l 1. It holds
!1 d nCj˛j ˛ A D f L1 .B/ ˛Š 1
0
X ˛ C@ D f
L1 .B/
j˛jDl
3
A nd lCn 5
n
2n
2 n 2
C1
k!kL1 .Rn /
DW B3 :
(2.61)
Proof. Use of Theorems 2.8, 2.11; (2.40), (2.21), and (2.31). We present Theorem 2.36. All as in Theorems 2.8, 2.11. Assume k!kL1 .Rn / < 1I p; q > 1 W ˛ 1 1 n p C q D 1, l > p ; D f 2 Lp .B/ for j˛j D l. Then for every x 2 B (almost every x 2 B, respectively), it holds 1 0 X n ˛ A k!kL .Rn / ncp d l p Cn DW A4 : (2.62) E .f / .x/ @ D f 1 Lp .B/
j˛jDl ˛
Additionally, assume that D f 2 Lp .B/ ; for 1 j˛j l 1. Then 0 0 11 2 ˛ D f X Lp .B/ j˛jC n CC B B 6 q d .f / .x/ 4cq;l;n @ @ AA ˛Š 1j˛jl1
0
X ˛ C@ D f j˛jDl
DW B4 :
Lp .B/
1 A ncp d
3 l pn Cn
5 k!kL
n 1 .R /
(2.63)
60
2 Multivariate Inequalities Based on Sobolev Representations
Proof. By Theorems 2.8, 2.11; (2.40), (2.24), and (2.33). Proposition 2.37. All as in Theorem 2.35. It holds (for every x 2 B and almost every x 2 B, respectively) 0 B .f / .x/ k!kL1 .B/ @
X
˛ d j˛j D f ˛Š
1j˛jl1
1
L1 .B/ C
A
1 0 n nd lCn 2 k!kL1 .Rn / X ˛ A @ C D f L1 .B/ 2n n2 C 1 j˛jDl
DW B5 :
(2.64)
Proof. By Theorem 2.35 and (2.48). We also have Theorem 2.38. Here all as in Corollary 2.12. Assume l n. Then for every x 2 B (almost every x 2 B, respectively), it holds ˇ ˇ Z ˇ ˇ ˇ f .y/ ! .y/ dy ˇˇ .f / .x/ WD ˇf .x/ B 1 0 X 1 ˛ ˛ @ Dy Œ.x y/ ! .y/ A kf kL1 .B/ 1 ˛Š 1j˛jl1
0
X ˛ C@ D f j˛jDl
1 L1 .B/
A k!kL
n 1 .R /
nd l :
(2.65)
Proof. Based on Corollary 2.12, Theorem 2.34 and (2.36).
˛ Theorem 2.39. Here all as in Corollary 2.12. Suppose ln and D f
L1 .B/
<1
for all ˛ W j˛j D lI f 2 L1 .B/. Then for every x 2 B (almost every x 2 B, respectively), we find 0 .f / .x/ @
X
1j˛jl1
1 1 ˛ A kf kL .B/ Dy Œ.x y/˛ ! .y/ 1 L1 .B/ ˛Š
1 0 n nd lCn 2 k!kL1 .Rn / X ˛ A: @ C D f L1 .B/ 2n n2 C 1 j˛jDl
Proof. Based on Corollary 2.12, Theorem 2.35 and (2.36).
(2.66)
2.3
Main Results
61
Theorem 2.40. Here all as in Corollary 2.12. Assume p; q > 1 W n ; p
˛
1 p
C
1 q
D 1,
l> D f 2 Lp .B/ for j˛j D l and f 2 Lp .B/. Then for every x 2 B (almost every x 2 B, respectively), we derive 0 1 X 1 ˛ ˛ A kf kL .B/ .f / .x/ @ Dy Œ.x y/ ! .y/ p Lq .B/ ˛Š 1j˛jl1
C ncp d
0
l pn Cn
X ˛ k!kL1 .Rn / @ D f
Lp .B/
j˛jDl
1 A:
(2.67)
Proof. Based on Corollary 2.12, Theorem 2.36 and (2.36). We also give Theorem 2.41. Here all as in Corollary 2.13 with Qˇl1 f .x/ as in (2.37). Let l n C jˇj. Then for every x 2 B (almost every x 2 B, respectively) it holds ˇ ˇ Z ˇ ˇ ˇ ˇ D ! .y/ f .y/ dy Qˇl1 f .x/ˇˇ Eˇ .f / .x/ WD ˇˇ D f .x/ .1/jˇj B 1 0 X ˛ A k!k1 nd ljˇj DW A5 : (2.68) @ D f L1 .B/
j˛jDl;˛ˇ
Also, we derive
ˇ ˇ Z ˇ ˇ ˇ ˇ jˇj ˇ D ! .y/ f .y/ dy ˇˇ ˇ .f / .x/ WD ˇ D f .x/ .1/ B 1 0 X 1 ˛Cˇ @ Dy Œ.x y/˛ ! .y/ A kf kL1 .B/ 1 ˛Š 1j˛jljˇj1
0
C@
X
1
˛ D f
L1 .B/
j˛jDl;˛ˇ
A k!k1 nd ljˇj :
(2.69)
Proof. By Corollary 2.13, (2.25), (2.39), and (2.42). We continue with Theorem 2.42. Here all as in Corollary 2.13. Assume l n C jˇj I ˛ < 1, all ˛ W j˛j D l; ˛ ˇ. Then for every x 2 B (almost D f L1 .B/
every x 2 B, respectively) we find 0 X ˛ Eˇ .f / .x/ @ D f j˛jDl;˛ˇ
1 L1 .B/
A k!k1
n
n 2 d ljˇjCn DW A6 : 2n n2 C 1 (2.70)
62
2 Multivariate Inequalities Based on Sobolev Representations
Additionally, assume f 2 L1 .B/. It holds 0
X
ˇ .f / .x/ @
1j˛jljˇj1
0
˛ D f
X
C@
1 1 ˛Cˇ A kf kL .B/ Dy Œ.x y/˛ ! .y/ 1 L1 .B/ ˛Š
j˛jDl;˛ˇ
1
n
2 A k!k1 n d ljˇjCn : n L1 .B/ 2n 2 C 1
(2.71)
Proof. By Corollary 2.13, (2.26), (2.39), and (2.42). We finish Ostrowski-type inequalities with Theorem 2.43. Here all as in Corollary 2.13. Assume p; q > 1 W ˛
D f 2 Lp .B/ all ˛ W j˛j D l; ˛ ˇ; l > every x 2 B, respectively), we derive 0 Eˇ .f / .x/ @
X
˛ D f
C
1 q
D 1,
C jˇj. Then for every x 2 B (almost
1 n
Lp .B/
j˛jDl;˛ˇ
n p
1 p
A k!k1 ncp d ljˇj p Cn DW A7 :
(2.72)
Additionally assume f 2 Lp .B/. It holds 0
X
ˇ .f / .x/ @
1j˛jljˇj1
0
C@
X
j˛jDl;˛ˇ
1 1 ˛Cˇ A kf kL .B/ Dy Œ.x y/˛ ! .y/ p Lq .B/ ˛Š
˛ D f
1 n
Lp .B/
A k!k1 ncp d ljˇj p Cn :
(2.73)
Proof. By Corollary 2.13, (2.27), (2.39), and (2.42). We make Remark 2.44. In preparation to present comparison of integral means inequalities we consider the open ball B1 D B1 .y0 ; r1 / B. We consider also a weight function 0 which is Lebesgue integrable on Rn with supp B1 ˝, R R and Rn .x/ dx D 1. Clearly here B1 .x/ dx D 1. For example for x2 B1; 1 .x/ WD Vol.B , 0 elsewhere, etc. 1/ We will apply the following principle. In general a constraint of the form jF .x/ Gj ˇ ˇ and R ", where F is a function G; " real numbers that all make sense, implies that ˇ Rn F .x/ .x/ dx G ˇ ". Next we give a series of comparison of integral means inequalities based on Ostrowski-type inequalities presented in this chapter. We use Remark 2.44.
2.3
Main Results
63
Theorem 2.45. All as in Theorem 2.33. Then ˇZ Z ˇ ˇ M .f / WD ˇ f .x/ .x/ dx f .x/ ! .x/ dx B1
B
Z Q
l1
ˇ ˇ .x/ dx ˇˇ A1 ;
(2.74)
ˇ ˇ f .x/ ! .x/ dx ˇˇ B1 :
(2.75)
f .x/
B1
and
ˇZ ˇ m .f / WD ˇˇ
Z f .x/
.x/ dx
B1
B
Theorem 2.46. All as in Theorem 2.34. Then M .f / A2 ;
(2.76)
m .f / B2 :
(2.77)
and
Theorem 2.47. All as in Theorem 2.35. Then M .f / A3 ;
(2.78)
m .f / B3 :
(2.79)
and
Theorem 2.48. All as in Theorem 2.36. Then M .f / A4 ;
(2.80)
m .f / B4 :
(2.81)
Theorem 2.49. All as in Proposition 2.37. Then m .f / B5 :
(2.82)
Theorem 2.50. All as in Theorem 2.41. Then ˇZ Z ˇ ˇ ˇ .x/ D f .x/ dx .1/jˇj D ! .x/ f .x/ dx M ˇ .f / WD ˇˇ B1
Z
B1
ˇ ˇ .x/ Qˇl1 f .x/ dx ˇˇ A5 :
B
(2.83)
64
2 Multivariate Inequalities Based on Sobolev Representations
Theorem 2.51. All as in Theorem 2.42. Then M ˇ .f / A6 :
(2.84)
We finish with Theorem 2.52. All as in Theorem 2.43. Then M ˇ .f / A7 :
(2.85)
2.4 Applications Example 2.53 (see also [5], p. 93). Let B WD fx 2 Rn W jx x0 j < g, and
' .x/ WD
8 ˆ <
2 !1 jxx0 j 1
e ˆ : 0,
; if jx x0 j < ; if jx x0 j :
(2.86)
R CallRc WD Rn ' .x/ dx > 0, then ˚ .x/ WD 1c ' .x/ 2 C01 .Rn / with supp˚ D B and Rn ˚ .x/ dx D 1 and max j˚j cons tan t n : We call ˚ a cut-off function. 1 One for this chapter’s results by choosing ! .x/ D ˚ .x/ or ! .x/ D Vol.B/ , etc., can give lots of applications. Here, selectively we give some special cases inequalities. We start with Chebyshev–Gr¨uss-type inequalities. Corollary 2.54 (to Theorem 2.26). All assumptions as in Theorem 2.26. Case of l D 1. Then ˇZ Z Z ˇ ˇ ˇ ˇ ! .x/ f .x/ g .x/ dx ! .x/ f .x/ dx ! .x/ g .x/ dx ˇˇ ˇ B
B
nd
k!k2L1 .Rn / 2
B
max ˛ kgk1;B kD ˛ f kmax 1;1;B C kf k1;B kD gk1;1;B : (2.87)
If f D g, then ˇZ Z 2 ˇˇ ˇ ˇ ˇ 2 ! .x/ f .x/ dx ˇ ˇ ! .x/ f .x/ dx ˇ B ˇ B nd k!k2L1 .Rn / kf k1;B kD ˛ f kmax 1;1;B :
(2.88)
References
65
Corollary 2.55 (to Theorem 2.27). All assumptions as in Theorem 2.27. Case of 1 f D g, l D n and ! .x/ WD Vol.B/ ; for all x 2 B, ! .x/ WD 0 on Rn B. Then ˇZ n Z 2 ˇˇ ˇ n C 1 2 ˇ ˇ 2 f .x/ dx ˇ ˇ f 2 .x/ dx n n ˇ B ˇ 2 d B 8 0 !19 n 2< = n j˛jn X 2 2 C1 d D ˛ f 4 kf kL .B/ @ A n w 1 .B/ L1 : ; ˛Š 2 1j˛jn1
113 X D ˛ f AA5 : C n @kf kL1 .B/ @ w L1 .B/ 0
0
(2.89)
j˛jDn
We continue the Ostrowski-type inequality. Corollary 2.56 (to Theorem 2.33). All as in Theorem 2.33. Case of l D 1. Then for every x 2 B it holds ˇ ˇ Z ˇ ˇ ˇf .x/ f .y/ ! .y/ dy ˇˇ nd k!kL1 .Rn / kD ˛ f kmax 1;1;B WD Z1 : ˇ
(2.90)
B
We finish chapter with a comparison of means inequality. Corollary 2.57 (to Corollary 2.56). All as in Corollary 2.56 and Remark 2.44. Then ˇZ ˇ Z ˇ ˇ ˇ f .x/ .x/ dx f .y/ ! .y/ dy ˇˇ Z1 : (2.91) ˇ B1
B
References 1. G.A. Anastassiou, Quantitative Approximations, Chapman & Hall/CRC, Boca Raton, New York, 2000. 2. G.A. Anastassiou, Probabilistic Inequalities, World Scientific, Singapore, New Jersey, 2010. 3. G.A. Anastassiou, Advanced Inequalities, World Scientific, Singapore, New Jersey, 2010. 4. G.A. Anastassiou, Multivariate Inequalities based on Sobolev Representations, Applicable Analysis, accepted 2011. 5. S. Brenner and L.R. Scott, The mathematical theory of finite element methods, Springer, N. York, 2008. 6. V. Burenkov, Sobolev spaces and domains, B.G. Teubner, Stuttgart, Leipzig, 1998. 7. P.L. Chebyshev, Sur les expressions approximatives des integrales definies par les autres prises entre les mˆemes limites, Proc. Math. Soc. Charkov, 2(1882), 93-98. h 1 Rb ¨ 8. G. Gr¨uss, Uber das Maximum des absoluten Betrages von a f .x/ g .x/ dx ba i Rb Rb 1 .ba/2 a f .x/ dx a g .x/ dx , Math. Z. 39 (1935), pp. 215-226. 9. A. Ostrowski, Uber die Absolutabweichung einer differentiabaren Funcktion von ihrem Integralmittelwert, Comment. Math. Helv. 10(1938), 226-227.
