Discrete Hilbert-Type Inequalities

161 Pages·2014·2.448 MB·English

by  Bicheng Yang

Discrete Hilbert-Type Inequalities

BY
Bicheng Yang
Department of Mathematics, Guangdong Education Institute, Guangdong Guangzhou,
510303,
People Republic of China
E-mail: [email protected]
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CONTENTS
Foreword i
Preface ii
CHAPTERS
1. Introduction 1
2. Improvements of Euler –Maclaurin’s Summation Formula: Preliminary
Theorems 14
3. Hilbert-Type Inequalities with the Homogeneous Kernel of Degree -1 27
4. Hilbert-Type Inequalities with the General Homogeneous Kernel 54
5. Some Innovative Hilbert-Type Inequalities 90
6. Multiple Hilbert-Type Inequalities 122
Index 153
i
FOREWORD
In 1934 G. H. Hardy, J. E. Littlewood and G. Polya published the book entitled "Inequalities", in which a few
theorems about Hilbert-type inequalities with the homogeneous kernels of degree -1 and some special cases were
considered. In 1991, Lizhi Xu first raised the way of weight coefficients to make a strengthened version of
Hilbert’s inequality. In 1998, by applying the way of weight coefficients and introducing an independent
parameter and the beta function, a best extension of Hilbert’s integral inequality was published in Journal of
Mathematical Analysis and Applications by the author. In 2004 the author added two pairs of conjugate exponents
and provided an extended Hilbert’s integral inequality. In 2009, the author published by Science Press (China) his
book entitled "The Norm of Operator and Hilbert-Type Inequalities". In that book, two large classes of Hilbert-
type inequalities with the homogeneous kernels of a negative number degree, including integrals and series were
discussed. However the author did not consider it an important case of Hardy-type inequalities. In October 2009,
the author published an e-book entitled "Hilbert-Type Integral Inequalities" by Bentham Science Publishers Ltd.
In that book, the author studied the broader case of Hilbert-type integral inequalities with the homogeneous
kernels of a real number degree, which provide the best recent extensions of corresponding results. A number of
equivalent forms as well as their reverses with several extended multiple inequalities in a number of particular
cases are considered.
The book is divided into six chapters. In Chapter 1, some preliminary materials on the theory and methods of
Hilbert-type inequalities, including the classical Hilbert’s inequality are discussed. Chapter 2 deals with an
optimization of the methods of estimating the series and the weight coefficients. Some introductory theorems of
improving the methods of Euler-Maclaurin summation formula are analyzed. In Chapter 3, by using the way of
weight coefficients some fundamental theorems and corollaries on the discrete Hilbert-type inequalities with the
homogeneous kernel of degree -1 are provided. The proofs regarding the best possible property of the constant
factors are left to be studied in Chapter 4. In Chapter 4, some discrete Hilbert-type inequalities and their reverses
with the general homogeneous kernel and the best constant factors are considered. These provide extensions of
certain results of Chapter 3. By applying the improved Euler-Maclaurin summation formula and notions from
Real Analysis, some particular examples are given. In Chapter 5 based upon some theorems of Chapter 4 and by
applying techniques from Real Analysis, the author has explained how to use particular parameters to formulate
some new Hilbert-type inequalities and their reverses with the best constant factors. In addition, a class of Hilbert-
type inequalities with the general measurable kernels is considered. In Chapter 6, the author has decently
formulated some lemmas and obtained two equivalent multiple Hilbert-type inequalities and their reverses with
the homogeneous kernel of a real number degree. These inequalities are the best extensions of the corresponding
inequalities in Chapter 4. Some special examples are also studied.
The author has succeeded to present in this book an extensive account of several Hilbert-type inequalities in a self
contained and rigorous manner. The book will be very useful not only to graduate students who study inequalities
in the broader domain of Mathematical Analysis but also to research mathematicians who need the latest
information to refer in.
Themistocles M. Rassias
National Technical University of Athens
Athens
Greece
ii
PREFACE
One Hundred years ago, in 1908, H. Wely published the well known Hilbert’s inequality. In 1925, G. H. Hardy
gave a best extension of it by introducing one pair of conjugate exponents (p, q), named as Hardy-Hilbert’s
inequality. The Hilbert-type inequalities are a more wide class of analysis inequalities which are with the bilinear
kernels, including Hardy-Hilbert’s inequality as the particular case. These inequalities are important in analysis
and its applications. By making a great effort of mathematicians in the world at about one hundred years, the
theory of Hilbert-type inequalities has now come into being. This book is a monograph about the theory of
discrete Hilbert-type inequalities with the homogeneous kernels of real number-degree and its applications. Using
the methods of series summation, Real Analysis, Functional Analysis and Operator Theory, and following the way
of weight coefficients, the author introduces a few independent parameters to establish a number of discrete
Hilbert-type inequalities with the homogeneous kernels of real number-degree and the best constant factors,
including some multiple inequalities. The equivalent forms and the reverses with the best constant factors are also
considered. As application, the author also considers some discrete Hilbert-type inequalities with the non-
homogeneous kernels and a large number of particular examples.
For reading and understanding this book, readers should hold the basic knowledge of real analysis and functional
analysis. This book is suited to the people who are interested in the fields of analysis inequalities and real analysis.
The author expects this book to help many readers to make good progress in research for discrete Hilbert-type
inequalities and their applications.
Bicheng Yang
Guangzhou, Guangdong,
P. R. China
Discrete Hilbert-Type Inequalities, 2011, 1-13 1
CHAPTER 1
Introduction
Abstract: In this chapter, we introduce some evolvements for the theory and methods of Hilbert-type inequalities,
including Hilbert’s inequality. We must emphasize some excellent works on discrete Hilbert-type inequalities and
Hilbert-type operators with multi-parameters in recent years, which have made more developments in this context. This
chapter will enhance the understanding of the readers of the content of the following several chapters.

1.1. HILBERT’S INEQUALITIES AND any a{a } l2 , there exists a
m m1
HILBERT’S OPERATOR
c{c } l2, satisfying
n n1
1.1.1. RESEARCH BACKGROUND OF 
c :(Ta)(n) am , nN (1.1.3)
HILBERT’S INEQUALITIES AND HILBERT’S n mn
m1
OPERATOR
(N is the set of positive integers). Hence, for any

In 1908, H. Weyl published the following well known b{b } l2, we may indicate the inner product
n n1
Hilbert’s inequality (Weyl ID 1908) [1]: If of Taandbas follows:
{a } and {b } are real sequences, satisfying  
n n (Ta,b)(c,b)( am )b
0  a2  and 0 b2  , then we mn n
n1 n n1 n n1 m1
have  
 1 a b . (1.1.4)
    mn m n
 1 a b (a2b2)12, (1.1.1) n1m1
mn m n n n
n1m1 n1 n1 Expressing the norm ofaas ||a|| ( a2)12, in
where the constant factor  is the best possible. We 2 n1 n
view of (1.1.4), inequality (1.1.1) may be rewritten as
named (1.1.1) as Hilbert’s inequality. The best possible
property of the constant factor  was proved by Schur (Ta,b)||a|| ||b|| , (1.1.5)
2 2
in 1911 (Schur JM 1991) [2]. He also gave the where||a|| ,||b|| 0. We may prove that T is a
following integral analogue of (1.1.1) at the same time: 2 2
If f(x) and g(x) are measurable functions, bounded operator and obtain the norm as||T ||
(Wilhelm AJM 1950) [7]. We callT Hilbert’s operator
 
0  f 2(x)dx   and 0  g2(x)dx  , with the kernel 1 . For||a|| 0, the equivalent
0 0 mn 2
then we get form of (1.1.5) is given as ||Ta|| ||a|| , e.t.
2 2
 
  1 f(x)g(y)dxdy   
0 0 xy ( am )2 2a2 , (1.1.6)
( f 2(x)dxg2(x)dx)12 , (1.1.2) n1 m1mn n1 n
0 0 where the constant factor 2is still the best possible.
where the constant factor  is still the best
Obviously, inequalities (1.1.6) and (1.1.1) are
possible. We call (1.1.2) as Hilbert’s integral
equivalent (Hardy CUP 1934) [3].
inequality. Inequalities (1.1.1) and (1.1.2) are
important in analysis and applications. We can see a
Similarly, if L2(0,) is a real function space, we
number of improvements and extensions in vast
mathematics literature, especially in the books (Hardy may define Hilbert’s integral operator as
~
CMP 1934) [3], (Mitrinovic KAP 1991) [4], (Kuang T :L2(0,) L2(0,), for any f L2(0,),
SSTP 2004) [5], (Hu WUP 2007) [6].
there exists a hTf L2(0,), satisfying
We may express inequality (1.1.1) by using the form (Tf)(y)h(y):  f(x)dx,y(0,). (1.1.7)
of operator as follows: If l2 is a space of real 0 xy
sequences, and T :l2 l2 is a linear operator, for Hence for anygL2(0,), we may still indicate the

inner product of Tf andgas follows:
Bicheng Yang
All rights reserved - © 2011 Bentham Science Publishers Ltd.
2 Discrete Hilbert-Type Inequalities Bicheng Yang
   For 01, in 1936, Ingham gave (Ingham JLMC
(Tf,g) ( 1 f(x)dx)g(y)dy
0 0 xy 1936)[9]: If 1 , then we have
2
 
  1 f(x)g(y)dxdy. (1.1.8)   
0 0 xy  aman a2 ; (1.1.15)
mn n
Setting the norm of f as|| f || ( f 2(x)dx)12, if n0m0 n0
2 0 if0 1 , then we get
2
|| f || ,||g|| 0, then (1.1.2) may be rewritten as
2 2   
~  aman   a2 . (1.1.16)
(Tf,g)|| f || || g || . (1.1.9) mn sin() n
~ 2 2 n0m0 n0
We have ||T || (Carleman U 1923) [8] and It is interesting that if we set x X ,y Y ,
2 2

inequality||Tf || || f || , which may be rewritten F(X) f(X )andG(Y) g(Y )
2 2 2 2
to the equivalent form of (1.1.2) as follows (Hardy (R, R is the set of real numbers) in (1.1.2), then we
CUP 1934) [3]:
obtain
  
 ( f(x)dx)2dy 2 f 2(x)dx, (1.1.10)  
xy   1 F(X)G(Y)dXdY
0 0 0   XY
where the constant factor 2is still the best possible. 2 2
It is obvious that inequality (1.1.10) is the integral ( F2(X)dX G2(X)dX)12. (1.1.17)
analogue of (1.1.6). 2 2

1.1.2. THE MORE ACCURATE HILBERT’S It is said that for 1 , inequality (1.1.17) is an
2
INEQUALITY
integral analogue of (1.1.5) (for G  F ) and

for0 1 , inequality (1.1.17) is not an integral
If we set the subscripts m, n of the double series from 0 2
to infinity, then we may rewrite inequality (1.1.1) analogue of (1.1.16), since two constant factors are
equivalently in the following form: different.
   
 ambn (a2b2)12, (1.1.11) By using the improved Euler-Maclaurin summation
mn2 n n
n0m0 n0 n0 formula and introducing one parameter, a few authors
where the constant factor  is still the best possible. gave some more accurate Hilbert-type inequalities as
Obviously, we may raise the following question: Is (1.1.13) (see (Yang MIA 2004) [10], (Yang JXNU
there a positive constant (2) , that makes 2005)[11], (Yang CM 2005)[12], (Yang AMS 2006)
[13], (Yang JM 2007)[14], (Yang AMS 2007)[15],
inequality (1.1.11) still valid as we replace 2 by in
(Yang IMF 2007)[16], (Zhong JZU 2008)[17]).
the kernel 1 ? The answer is positive. That is the
mn2
following more accurate Hilbert’s inequality (for short, 1.1.3. HILBERT’S INEQUALITY WITH ONE
Hilbert’s inequality) (Hardy CUP 1934) [3]: PAIR OF CONJUGATE EXPONENTS
   
 ambn (a2b2)12 , (1.1.12) In 1925, by introducing one pair of conjugate
mn1 n n
n0m0 n0 n0 exponents (p,q)(1  1 1), Hardy and Riesz gave
where the constant factoris the best possible. Since p q
an extension of (1.1.1) as follows (Hardy PLMS 1925)
fora ,b 0,1, we find
m n [18]: If p1, a ,b 0 , such that
    n n
mamnbn mamnbn1, 0 ap  and 0 bq  , then we
n0m0 n0m0 n1 n n1 n
then by (1.1.12) and for1, we have obtain
   
  ambn ( a2 b2)12. (1.1.13) ammbnn  sin()(anp)1p(bnq)q1, (1.1.18)
mn n n n1m1 p n1 n1
n0m0 n0 n0
For1 2, inequality (1.1.13) is a refinement of where the constant factorsin(/p)is the best possible.
(1.1.11), which is equivalently a refinement of (1.1.1). The equivalent form of (1.1.18) is as follows:
Obviously, we have a refinement of (1.1.6), which is   
equivalent to (1.1.13) as follows: ( am )p [  ]pap , (1.1.19)
mn sin(/p) n
   n1 m1 n1
( am )2 2a2(12). (1.1.14)
mn n
n0 m0 n0
Introduction Discrete Hilbert-Type Inequalities 3
where the constant factor [  ]p is still the best T :Lp(0,)Lp(0,)as: for any f(0)
sin(/p) p
possible. In the same way, inequalities (1.1.12) and Lp(0,) , there exists a hT f Lp(0,) ,
(1.1.14) (for1) may be extended to the following p
satisfying
equivalent forms (Hardy CUP 1934) [3]:
  ambn   ( ap)1p( bq)q1, (1.1.20) (Tp f)(y)h(y):0 x1yf(x)dx,
mn1 sin() n n
n0m0 p n0 n0 y(0,). (1.1.27)
 ( am )p [  ]p ap , (1.1.21) And for anyg(0)Lq(0,), we can indicate the
mn1 sin(/p) n 
n0 m0 n0 formal inner product ofT f andgas follows:
p
where the constant factors  and[  ]p are
sin(/p) sin(/p) (T f,g) 1 f(x)g(y)dxdy. (1.1.28)
the best possible. And the equivalent integral p xy
0 0
analogues of (1.1.18) and (1.1.19) are given as follows: Setting the norm of f as|| f || ( f p(x)dx)1p ,
 1 f(x)g(y)dxdy p 0
0 0 xy then inequality (1.1.22) may be rewritten as follows
 sin(p)(0 f p(x)dx)1p(0gq(x)dx)1q, (1.1.22) (Tp f,g) sin(/p) || f ||p||g||q. (1.1.29)
( f(x)dx)pdy [  ]p f p(x)dx. If(p,q)is not one pair of conjugate exponents, we get
0 0 xy sin(p) 0 the following result (Hardy CUP 1934) [3]: If
(1.1.23) p 1,q 1,1  1 1,02 1  1 1, then
p q p q
We call (1.1.18) and (1.1.20) as Hardy-Hilbert’s
   
inequality and call (1.1.22) as Hardy-Hilbert’s integral  ambn  K(ap)1p(bq)1q , (1.1.30)
inequality. (mn) n n
n1m1 n1 n1
Inequality (1.1.20) may be expressed in the form of where K  K(p,q) relates to p,q ; only for
operator as follows: If lpis a space of real sequences, 1  1 1, 2 1  1 1 , the constant factor
p q p q
T :lp lp is a linear operator, such that for any
p K is the best possible. The integral analogues of
non-negative sequence a{a } lp, there exists (1.1.30) are given as follows:
m m1  
Tpa c{cn}n1lp, satisfying 0 0 (x1y)f(x)g(y)dxdy
c :(T a)(n) am , nN (1.1.24)  K( f p(x)dx)1p(gq(x)dx)1q. (1.1.31)
n p mn1 0 0 0
m0 We also find an extension of (1.1.31) as (Mitrinovic
(N =N {0} ). And for any non-negative CAP 1991) [4]: If p 1,q 1, 1  1 1,
0 p q
sequenceb{b } lq, we can indicate the formal 02 1  1 1, then we get
n n1 p q
inner product of T aandbas:  
p   f(x)g(y)dxdy k(p,q)
    |xy|
(Tpa,b)(mamn1)bn ( f p(x)dx)1p( gq(x)dx)1q. (1.1.32)
n0 m0  
  For f(x),g(x)0,x(,0], inequality (1.1.32)
 1 a b . (1.1.25)
mn1 m n reduces to (1.1.31). Levin also studied the expression
n0m0
forms of the constant factors in (1.1.30) and (1.1.31)
Setting the norm ofa as||a|| ( ap)1p , then (Levin JIMS 1937) [19], but he could not prove their best
p n0 n
possible property. In 1951, Bonsall considered the case of
inequality (1.1.20) may be rewritten as follows
(1.1.31) in the general kernel (Bonsall JMOS 1951) [20].
(T a,b)  ||a|| ||b|| , (1.1.26)
p sin(/p) p q
1.1.4. A HILBERT-TYPE INEQUALITY WITH
where||a||p,||b||q0. We call TpHardy-Hilbert’s THE HOMOMGENEOUS KERNEL OF
operator with the kernel 1 . DEGREE -1 AND SOME PARTICULAR
mn1 CASES

Similarly, ifLp(0,)is a real function space, we may If  R, the function k(x,y) is measurable in
define the following Hardy-Hilbert’s integral operator (0,)(0,) , satisfying for any x,y,u 0 ,
4 Discrete Hilbert-Type Inequalities Bicheng Yang
k(ux,uy)uk(x,y) , we name k(x,y) as the   ambn  pq( ap)1p( bq)1q,
homogeneous function of degree. max{m,n} n n
n1m1 n1 n1

(1.1.39)
Supposing that (p,q)(1  1 1) is one pair of
p q   
conjugate exponents with p1, k(x,y)0 is a ( am )p (pq)pap ; (1.1.40)
max{m,n} n
homogeneous function of degree -1 in n1 m1 n1


(0,)(0,) , if k   k(u,1)u1/pdu is finite, (3) for k(x,y) ln(x/y) in (1.1.33)-(1.1.36), they
0 xy
 1 deduce to the following two pairs of equivalent forms:
then we have k   k(1,v)v qdvand the following
0 ln(x/y) f(x)g(y)dxdy
equivalent inequalities (Hardy CUP 1934) [3]: xy
0 0
k(x,y)f(x)g(y)dxdy [  ]2( f p(x)dx)1p(gq(x)dx)1q , (1.1.41)
0 0 sin(p) 0 0
k( f p(x)dx)1p(gq(x)dx)1q, (1.1.33) [ln(x/y)f(x)dx]pdy
0 0 xy
0 0
  
 ( k(x,y)f(x)dx)pdy kp f p(x)dx, [  ]2p f p(x)dx; (1.1.42)
0 0 0 sin(p) 0
(1.1.34)
   
ln(m/n)ambn [  ]2(ap)1p(bq)1q,
where the constant factors k and kp are the best mn sin() n n
n1m1 p n1 n1
possible .
(1.1.43)

If both k(u,1)u1/p and k(1,u)u1/q are decreasing  [ ln(m/n)am]p [  ]2p ap . (1.1.44)
mn sin() n
functions in (0,) , then we have the following n1 m1 p n1
equivalent inequalities:
Note. The constant factors in the above inequalities are
   
k(m,n)a b k(ap)1p(bq)1q, all the best possible. We name (1.1.39) and (1.1.43) as
m n n n
Hardy–Littlewood–Polya’s inequalities, or for short,
n1m1 n1 n1
(1.1.35) H-L-P inequalities, and call (1.1.37) and (1.1.41) H-L-
P integral inequalities. We find that the kernels in the
  
(k(m,n)a )p kpap . (1.1.36) above inequalities are all decreasing, but this is not
m n
necessary. For example, we find the following two
n1 m1 n1
pairs of equivalent forms with the non-decreasing
 kernel (Yang JGEI 2006) [21], (Yang JJU 2007) [22]:
For0 p 1, ifk  k(u,1)u1/pduis finite, then
 
0   |ln(x/y)| f(x)g(y)dxdy
we have the reverses of (1.1.33) and (1.1.34) (Note: we 0 0 max{x,y}
have not seen any proof of (1.1.33) - (1.1.36) and the (p2 q2)( f p(x)dx)1p(gq(x)dx)1q ,
reverse particular cases in (Hardy CUP 1934) [3]).
0 0

(1.1.45)
We name k(x,y)the kernel of inequalities (1.1.33) -
 
(1.1.36). If all the integrals and series in the right hand  [ |ln(x/y)| f(x)dx]pdy
max{x,y}
0 0
side of inequalities (1.1.33)-(1.1.36) are positive, we

still obtain the following particular cases (Hardy CUP (p2 q2)p f p(x)dx; (1.1.46)
1934) [3]: (1) Fork(x,y) 1 in (1.1.33) -(1.1.36), 0
xy  
they deduce to (1.1.22), (1.1.23), (1.1.18) and (1.1.19);  |ln(m/n)| a b
max{m,n} m n
(2) for k(x,y) 1 in (1.1.33)-(1.1.36), they n1m1
max{x,y}  
deduce to the following two pairs of equivalent forms: (p2 q2)(ap)1p(bq)1q , (1.1.47)
n n
 1 f(x)g(y)dxdy n1 n1
0 0 max{x,y}   
( |ln(m/n)| a )p (p2 q2)pap ,
 pq( f p(x)dx)1p(gq(x)dx)1q , (1.1.37) max{m,n} m n
n1 m1 n1
0 0
(1.1.48)
  
 ( f(x) dx)pdy(pq)p f p(x)dx; where the constant factors p2 q2 and(p2 q2)p
max{x,y}
0 0 0
(1.1.38) are the best possible. We call (1.1.47) Hilbert-Yang’s