Recent Developments of Hilbert-Type Discrete and Integral Inequalities with Applications

This paper deals with recent developments of Hilbert-type discrete and integral inequalities by introducing kernels, weight functions, and multiparameters. Included are numerous generalizations, extensions, and refinements of Hilbert-type inequalities involving many special functions such as beta, gamma, logarithm, trigonometric, hyper-bolic, Bernoulli’s functions and Bernoulli’s numbers, Euler’s constant, zeta function, and hypergeometric functions with many applications. Special attention is given to many equivalent inequalities and to conditions under which the constant factors involved in inequalities are the best possible. Many particular cases of Hilbert-type inequalities are presented with numerous applications. A large number of major books and recent research papers published during 2009–2012 are included to stimulate new interest in future study and research.


Introduction
Historically, mathematical analysis has been the major and significant branch of mathematics for the last three centuries.Indeed, inequalities became the heart of mathematical analysis.Many great mathematicians have made significant contributions to many new developments of the subject, which led to the discovery of many new inequalities with proofs and useful applications in many fields of mathematical physics, pure and applied mathematics.Indeed, mathematical inequalities became an important branch of modern mathematics in International Journal of Mathematics and Mathematical Sciences the twentieth century through the pioneering work entitled Inequalities by G. H. Hardy, J. E. Littlewood, and G. P òlya, which was first published treatise in 1934.This unique publication represents a paradigm of precise logic, full of elegant inequalities with rigorous proofs and useful applications in mathematics.
During the twentieth century, discrete and integral inequalities played a fundamental role in mathematics and have a wide variety of applications in many areas of pure and applied mathematics.In particular, David Hilbert 1862-1943 first proved Hilbert's double series inequality without exact determination of the constant in his lectures on integral equations.If {a m } and {b n } are two real sequences such that 0 < ∞ m 1 a 2 m < ∞ and 0 < ∞ n 1 b 2 n < ∞, then the Hilbert's double series inequality is given by In 1908, Weyl 1 published a proof of Hilbert's inequality 1.1 , and in 1911, Schur 2 proved that π in 1.1 is the best possible constant and also discovered the integral analogue of 1.1 , which became known as the Hilbert's integral inequality in the form where f and g are measurable functions such that 0 < ∞ 0 f 2 x dx < ∞ and 0 < ∞ 0 g 2 y dy < ∞, and π in 1.2 is still the best possible constant factor.A large number of generalizations, extensions, and refinements of both 1.1 and 1.2 are available in the literature in Hardy et al. 3 ,Mitrinović et al. 4 , Kuang 5 , and Hu 6 .Considerable attention has been given to the well-known classical Hardy-Littlewood-Sobolev HLS inequality see Hardy et al. 3

in the form
R n R n f x g y x − y λ dx dy ≤ C n,λ,p f p g q , 1.3 for every f ∈ L p R n and g ∈ L q R n , where 0 < λ < n, 1 < p, q < ∞ such that 1/p 1/q λ/n 2, and f p is the L p R n norm of the function f.For arbitrary p and q, an estimate of the upper bound of the constant C n,λ,p was given by Hardy, Littlewood, and Sobolev, but no sharp value is known up till now.However, for the special case, p q 2n/ 2n − λ , the sharp value of the constant was found as In 1958, Stein and Weiss 7 generalized the double-weighted inequality of Hardy and Littlewood in the form with the same notation as in 1.3 : R n R n f x g x |x| α x − y λ y β dx dy ≤ C α,β,n,λ,p f p g q , 1.5 where α β ≥ 0, and the powers of α and β of the weights satisfy the following conditions 1 − 1/p − λ/n < α/n < 1 − 1/p , and 1/p 1/q 1/n λ α β 2. Inequality 1.5 and its proof given by Stein and Weiss 7 represent some major contribution to the subject.
On the other hand, Chen et al. 8 used weighted Hardy-Littlewood-Sobolev inequalities 1.3 and 1.5 to solve systems of integral equations.In 2011, Khotyakov 9 suggested two proofs of the sharp version of the HLS inequality 1.3 .The first proof is based on the invariance property of the inequality 1.3 , and the second proof uses some properties of the fast diffusion equation with the conditions λ n − 2, n ≥ 3 on the sharp HLS inequality 1.3 .
The main purpose of this paper is to describe recent developments of Hilbert's discrete and integral inequalities in different directions with many applications.Included are many generalizations, extensions, and refinements of Hilbert-type inequalities involving many special functions such as beta, gamma, logarithm, trigonometric, hyperbolic, Bernoulli's functions and Bernoulli's numbers, Euler's constant, zeta function, and hypergeometric functions.Special attention is given to many equivalent inequalities and to conditions under which constant factors involved inequalities are the best possible.Many particular cases of Hilbert-type inequalities are presented with applications.A large number of major books and recent research papers published during 2009-2012 are included in references to stimulate new interest in future study and research.

Operator Formulation of Hilbert's Inequality
Suppose that R is the set of real numbers and where N is the set of positive integers.Hence, for any sequence b {b n } ∞ n 1 ∈ l 2 , we define the inner product of Ta and b as follows: International Journal of Mathematics and Mathematical Sciences Using 2.2 , inequality 1.1 can be rewritten in the operator form where a 2 and b 2 > 0. It follows from Wilhelm 10 that T is a bounded operator and the norm T π and T is called Hilbert's operator with the kernel 1/ m n .For a 2 > 0, the equivalent form of 2.3 is given as Ta 2 < π a 2 , that is, where the constant factor π 2 is still the best possible.Obviously, inequality 2.4 and 1.1 are equivalent see Hardy et al. 3 .We may define Hilbert's integral operator as follows: Hence, for any g ∈ L 2 R , we may still define the inner product of Tf and g as follows: Tf, g Setting the norm of f as where the constant factor π 2 is still the best possible.It is obvious that inequality 2.8 is the integral analogue of 2.4 .

A More Accurate Discrete Hilbert's Inequality
If we set the subscripts m, n of the double series from zero to infinity, then, we may rewrite inequality 1.1 equivalently in the following form: where the constant factor π is still the best possible.Obviously, we may raise the following question: Is there a positive constant α < 2 , that makes inequality still valid as we replace 2 by α in the kernel 1/ m n 2 ?The answer is positive.That is, the following is more accurate Hilbert's inequality for short, Hilbert's inequality see Hardy et al. 3 : where the constant factor π is the best possible.Since for then, by 3.2 and for α ≥ 1, we obtain

3.4
For 1 ≤ α < 2, inequality 3.4 is a refinement of 3.1 .Obviously, we have a refinement of 2.4 , which is equivalent to 3.4 as follows: International Journal of Mathematics and Mathematical Sciences
Using the improved version of the Euler-Maclaurin summation formula and introducing new parameters, several authors including Yang see 13-15 recently obtained several more accurate Hilbert-type inequalities and some new Hardy-Hilbert inequality with applications.

Hilbert's Inequality with One Pair of Conjugate Exponents
In 1925, by introducing one pair of conjugate exponents p, q with 1/p 1/q 1, Hardy 16 gave an extension of 1.1 as follows: where the constant factor π/ sin π/p is the best possible.The equivalent discrete form of 4.1 is as follows: where the constant factor π/ sin π/p p is still the best possible.Similarly, inequalities 3.2 and 3.5 for α 1 may be extended to the following equivalent forms see Hardy et al. 3 : where the constant factors π/ sin π/p and π/ sin π/p p are the best possible.The equivalent integral analogues of 4.1 and 4.2 are given as follows: 4.6 4.1 and 4.3 as Hardy-Hilbert's inequality and call 4.5 as Hardy-Hilbert's integral inequality.Inequality 4.3 may be expressed in the form of operator as follows: T p : l p → l p is a linear operator, such that for any nonnegative sequence a {a m } ∞ m 1 ∈ l p , there exists And for any nonnegative sequence b {b n } ∞ n 1 ∈ l q , we can define the formal inner product of T p a and b as follows: Then inequality 4.3 may be rewritten in the operator form where a p , b q > 0. The operator T p is called Hardy-Hilbert's operator.
Similarly, we define the following Hardy-Hilbert's integral operator T p : L p R → L p R as follows: for any f ≥ 0 ∈ L p R , there exists an h T p f ∈ L p R , defined by And for any g ≥ 0 ∈ L q R , we can define the formal inner product of T p f and g as follows: Then inequality 4.5 may be rewritten as follows:

International Journal of Mathematics and Mathematical Sciences
On the other hand, if p, q is not a pair of conjugate exponents, then we have the following results see Hardy et where K K p, q relates to p, q, only for 1/p 1/q 1, λ 2 − 1/p 1/q 1, the constant factor K is the best possible.The integral analogue of 4.13 is given by We also find an extension of 4.14 as follows see Mitrinović et al. 4 : For f x g x 0, x ∈ −∞, 0 , inequality 4.15 reduces to 4.14 .Leven 17 also studied the expression forms of the constant factors in 4.13 and 4.14 .But he did not prove their best possible property.In 1951, Bonsall 18 considered the case of 4.14 for the general kernel.

A Hilbert-Type Inequality with the General Homogeneous
Kernel of Degree −1 du and the following equivalent integral inequalities: where the constant factor k is the best possible and k 1 1, u u −1/q are decreasing in R , then we have the following equivalent discrete forms: 2 if k 1 x, y 1/ max{x, y} in 5.1 -5.4 , they reduce the following two pairs of equivalent forms: International Journal of Mathematics and Mathematical Sciences 3 if k 1 x, y ln x/y / x − y in 5.1 -5.4 , they reduce to the following two pairs of equivalent inequalities: 5.12 Note 3. The constant factors in the above inequalities are all the best possible.We call 5.7 and 5.11 Hardy-Littlewood-Pólya's inequalities or H-L-P inequalities .We find that the kernels in the above inequalities are all decreasing functions.But this is not necessary.For example, we find the following two pairs of equivalent forms with the nondecreasing kernel see Yang 19 : where the constant factors p 2 q 2 and p 2 q 2 p are the best possible.Another type inequalities with the best constant factors are as follows see Xin and Yang 20 : where the constant factor c 0 p

Two Multiple Hilbert-Type Inequalities with the Homogeneous Kernels of Degree −n 1
Suppose n ∈ N \ {1}, n numbers p, q, . .., r satisfying p, q, . . ., r > 1, all decreasing functions with respect to any single variable in R , then, we have International Journal of Mathematics and Mathematical Sciences Note 4. The authors did not write and prove that the constant factor k in the above inequalities is the best possible.For two numbers p and q n 2 , inequalities 6.2 and 6.3 reduce, respectively, to 5.1 and 5.3 .

Modern Research for Hilbert's Integral Inequality
In 1979, based on an improvement of H ölder's inequality, Hu 21 proved a refinement of 1.2 for f g as follows: where B u, v is the beta function.
In 1999, Kuang 28 gave another extension of 1.2 as follows:

7.4
We can refer to the other works of Kuang in 5, 29 .In 1999, using the methods of algebra and analysis, Gao 30 proved an improvement of 1.2 as follows: where

e y
∞ 0 e x / x y dx.We also refer to works of Gao and Hsu in 31 .

International Journal of Mathematics and Mathematical Sciences 13
In 2002, using the operator theory, Zhang 32 gave an improvement of 1.2 as follows: 7.6

On the Way of Weight Coefficient for Giving a Strengthened Version of Hilbert's Inequality
In 1991, for making an improvement of 1.1 , Hsu and Wang 33 raised the way of weight coefficient as follows: at first, using Cauchy's inequality in the left-hand side of 1.1 , it follows that

8.1
Then, we define the weight coefficient and rewrite 8.1 as follows:

8.3
Setting where θ n π − ω n n 1/2 , and estimating the series of θ n , it follows that International Journal of Mathematics and Mathematical Sciences Thus, result 8.4 yields In view of 8.3 , a strengthened version of 1.1 is given by In 1997, using the way of weight coefficient and the improved Euler-Maclaurin's summation formula, Yang and Gao 36, 37 showed that where 1 − γ 0.42278433 γ is the Euler constant .
In 1998, Yang and Debnath 38 gave another strengthened version of 2.6 , which is an improvement of 8.8 .We can also refer to some strengthened versions of 3.2 and 4.3 in papers of Yang 39 and Yang and Debnath 40 .

Hilbert's Inequality with Independent Parameters
In 1998, using the optimized weight coefficients and introducing an independent parameter λ ∈ 0, 1 , Yang 27 provided an extension of 1.2 as follows.
If 0 where the constant factor B λ/2, λ/2 is the best possible.The proof of the best possible property of the constant factor was given by Yang 41 , and the expressions of the beta function B u, v are given in Wang and Guo 42 : 9.2 Some extensions of 4.1 , 4.3 , and 4.5 were given by Yang and Debnath 43-45 as follows.
where the constant factor B p λ − 2 /p, q λ − 2 /q is the best possible.Yang International Journal of Mathematics and Mathematical Sciences In 2004, Yang 49 proved the following dual form of 4.1 : q n 1/q .9.8 Inequality 9.8 reduces to 4.1 when p q 2. For λ 1, 9.7 reduces to the dual form of 4.3 as follows: q n 1/q .9.9 We can find some extensions of the H-L-P inequalities with the best constant factors such as 5.5 -5.16 see 13, 50, 51 by introducing some independent parameters.In 2001, by introducing some parameters, Hong 52 gave a multiple integral inequality, which is an extension of 4.1 .He et al. 53 gave a similar result for particular conjugate exponents.For making an improvement of their works, Yang 54 gave the following inequality, which is a best extension of 4.
where the constant factor 1/Γ λ n i 1 p i λ − n /p i is the best possible.In particular, for

9.11
In 2003, Yang and Rassias 55 introduced the way of weight coefficient and considered its applications to Hilbert-type inequalities.They summarized how to use the way of weight coefficient to obtain some new improvements and generalizations of the Hilbert-type inequalities.Since then, a number of authors discussed this problem see 56-77 .But how to give a uniform extension of inequalities 9.8 and 4.1 with a best possible constant factor was solved in 2004 by introducing two pairs of conjugate exponents.

Hilbert-Type Inequalities with Multiparameters
In 2004, by introducing an independent parameter λ > 0 and two pairs of conjugate exponents p, q and r, s with 1/p 1/q 1/r 1/s 1, Yang 78 gave an extension of 1.2 as follows: if p, r > 1 and the integrals of the right-hand side are positive, then where the constant factor π/λ sin π/r is the best possible.For λ 1, r q, s p, inequality 10.1 reduces to 4.5 ; for λ 1, r p, s q, inequality 10.1 reduces to the dual form of 4.5 as follows: In 2005, by introducing an independent parameter λ > 0, and two pairs of generalized conjugate exponents p 1 , p 2 , . . ., p n and r 1 , r 2 , . . ., r n with n i 1 1/p i n i 1 1/r i 1, Yang et al. 79 gave a multiple integral inequality as follows: where the constant factor 1/Γ λ n i 1 λ/r i is the best possible.For n 2, p 1 p, p 2 q, r 1 r, and r 2 s, inequality 10.3 reduces to the following: It is obvious that inequality 10.4 is another best extension of 4.5 .

International Journal of Mathematics and Mathematical Sciences
In 2006, using two pairs of conjugate exponents p, q and r, s with p, r > 1, Hong 80 gave a multivariable integral inequality as follows. If where the constant factor Γ n 1/α / βα n−1 Γ n/α B λ/r, λ/S is the best possible.In particular, for n 1, 10.5 reduces to Hong's work in 81 ; for n β 1, 10.5 reduces to 10.4 .In 2007, Zhong and Yang 82 generalized 10.5 to a general homogeneous kernel and proposed the reversion.
We can find another inequality with two parameters as follows see Yang 83 : where α, λ > 0, αλ ≤ min{r, s}.In particular, for α 1, we have For λ 1, r q, inequality 10.7 reduces to 4.1 , and for λ 1, r p, 10.7 reduces to 9.8 .Also we can obtain the reverse form as follows see Yang 84 : where 0 < p < 1, 1/p 1/q 1.The other results on the reverse of the Hilbert-type inequalities are found in Xi 85 and Yang 86 .In 2006, Xin 87 gave a best extension of H-L-P integral inequality 5.10 as follows: x q 1− λ/s −1 g q x dx 1/q .10.9 In 2007, Zhong and Yang 88 gave an extension of another H-L-P integral inequality 5.5 as follows: Zhong and Yang 89 also gave the reverse form of 10.10 .Considering a particular kernel, Yang 90 proved Yang 91 also proved that Using the residue theory, Yang 92 obtained the following inequality: where

Operator Expressions of Hilbert-Type Inequalities
Suppose that H is a separable Hilbert space and T : H → H is a bounded self-adjoint semipositive definite operator.In 2002, Zhang 32 proved the following inequality: where a, b is the inner product of a and b, and a a, a is the norm of a. Since the Hilbert integral operator T defined by 2.5 satisfies the condition of 11.1 with T π, then inequality 1.2 may be improved as 7.6 .Since the operator T p defined by 4.7 for p q 2 satisfies the condition of 11.1 see Wilhelm 10 , we may improve 3.2 to the following form: The key of applying 11.1 is to obtain the norm of the operator and and to show the semidefinite property.Now, we consider the concept and the properties of Hilbert-type integral operator as follows.
Suppose that p > 1, 1/p 1/q 1, L r R r p, q are real normal linear spaces and k x, y is a nonnegative symmetric measurable function in R 2 satisfying We define an integral operator as for any f ≥ 0 ∈ L p R , there exists h Tf ∈ L p R , such that Tf y h y : Or, for any g ≥ 0 ∈ L q R , there exists h Tg ∈ L q R , such that In 2006, Yang 98 proved that the operator T defined by 11.5 or 11.6 are bounded with T ≤ k 0 p .The following are some results in this paper: If ε > 0 is small enough and the integral ∞ 0 k x, t x/t 1 ε /r dt r p, q; x > 0 is convergent to a constant k ε p independent of x satisfying k ε p k 0 p o 1 ε → 0 , then T k 0 p .If T > 0, f ∈ L p R , g ∈ L q R , f p , g q > 0, then we have the following equivalent inequalities: Tf, g < T • f p g q , 11.7 Tf p < T • f p .11.8 Some particular cases are considered in this paper.Yang 99 also considered some properties of Hilbert-type integral operator for p q 2 .For the homogeneous kernel of degree −1, Yang 100 found some sufficient conditions to obtain T k 0 p .We can see some properties of the discrete Hilbert-type operator in the discrete space in Yang 101-104 .Recently, Bényi and Oh 105 proved some new results concerning best constants for certain multilinear integral operators.In 2009, Yang 106 summarized the above part results.Some other works about Hilbert-type operators and inequalities with the general homogeneous kernel and multiparameters were provided by several other authors see 107-114 .

Some Basic Hilbert-Type Inequalities
If the Hilbert-type inequality relates to a single symmetric homogeneous kernel of degree −1 such as 1/ x y or | ln x/y |/ x y and the best constant factor is a more brief form, which does not relate to any conjugate exponents such as 1.2 , then we call it basic Hilbert-type integral inequality.Its series analogue if exists is also called basic Hilbert-type inequality.If the simple homogeneous kernel is of degree −λ λ > 0 with a parameter λ and the inequality cannot be obtained by a simple transform to a basic Hilbert-type integral inequality, then we call it a basic Hilbert-type integral inequality with a parameter.
For examples, we call the following integral inequality, that is, 1.2 as and the following H-L-P inequalities for p 2 in 5.5 and 5.10 : 12.7 Similar to discrete inequality 3.7 , the following inequality: 12.8 is called basic Hilbert-type integral inequality with a parameter λ ∈ 0, ∞ .Also we find the following basic Hilbert-type inequalities: 12.11 It follows from 5.15 with p q 2 that

International Journal of Mathematics and Mathematical Sciences
Using the way of weight functions and the techniques of discrete and integral-type inequalities with some additional conditions on the kernel, a half-discrete Hilbert-type inequality with a general homogeneous kernel of degree −λ ∈ R is obtained as follows: k λ x, n a n dx < k λ 1 f p,φ a q,ψ , 13.3 where k λ 1 ∞ 0 k λ t, 1 t λ 1 −1 dt ∈ R .This is an extension of the above particular result with the best constant factor k λ 1 see Yang and Chen 133 .If the corresponding integral inequality of a half-discrete inequality is a basic Hilbert-type integral inequality, then we call it the basic half-discrete Hilbert-type inequality.Substituting some particular kernels in the main result found in 133 leads to some basic half-discrete Hilbert-type inequalities as follows: satisfying for any x, y, u > 0, k ux, uy u α k x, y , then k x, y is called the homogeneous function of degree α.In 1934, Hardy et al. 3 published the following theorem: suppose

Note 2 .
finite, then we have the reverses of 5.1 and 5.2 .We have not seen any proof of 5.1 -5.4 and the reverse examples in 3 .We call k 1 x, y the kernel of 5.1 and 5.2 .If all the integrals and series in the righthand side of inequalities 5.1 -5.4 are positive, then we can obtain the following particular examples see Hardy et al. 3 : 1 for k 1 x, y 1/ x y in 5.1 -5.4 , they reduce to 4.5 , 4.6 , 4.1 , and 4.2 ;

2 1/ 2 . 7 . 1 a − x f 2 x dx b 0 b − x g 2 y dy 1 / 2 . 7 . 2
Since then, Hu 6 published many interesting results similar to 7.1 .In 1998, Pachpatte 22 gave an inequality similar to 1.2 as follows: for a, b > 0, Some improvements and extensions have been made by Zhao and Debnath 23 , Lu 24 , and He and Li 25 .We can also refer to other works of Pachpatte in 26 .In 1998, by introducing parameters λ ∈ 0, 1 and a, b ∈ R a < b , Yang 27 gave an extension of 1.2 as follows: a, b, c > 0 .The constant factors in the above new inequalities are all the best possible.Some other new results are proved by several authors see75, 93-97 .

8 ∞ 0 f 2 x dx ∞ 0 g 2 x dx 1 / 2 . 12 . 4 International 2 , 12 . 6 where c 0 8 ∞ n 1
type integral inequalities.In 2006, Yang 115 gave the following basic Hilberttype integral inequality: ∞ 0 ∞ 0 ln x/y f x g y max x, y dx dy < Journal of Mathematics and Mathematical Sciences In 2011, Yang 116 gave the following basic Hilbert-type integral inequalities: −1 n / 2n − 1 2 7.3277 .In 2005, Yang 115, 117-120 gave a basic Hilbert-type integral inequality with a parameter λ ∈ 0, 1 : recent years, Pogány et al. 121-125 made some new important contributions to discrete Hilbert-type inequalities with nonhomogeneous kernels using special functions.On the other hand, in 2006-2011, Xie et al. 75, 95-97, 126, 127 have investigated many Hilberttype integral inequalities with the particular kernels such as |x y| −λ similar to inequality 4.15 .In 2009-2012, Yang 128, 129 considered the compositions of two discrete Hilberttype operators with two conjugate exponents and kernels m n , and in 2010, Liu and Yang 108 also studied the compositions of two Hilbert-type integral operators with the general homogeneous kernel of negative degree and obtained some new results with applications.Hardy et al. 3 proved some results in Theorem 351 without any proof of the constant factors as best possible.Yang 130 introduced an interval variable to prove that the constant factor is the best possible.In 2011, Yang 131 proved the following half-discrete Hilbert-type inequalities with the best possible constant factor B λ 1 , λ 2 : 1/p < ∞} are real normal spaces with the norms ||a|| p and ||f|| p .We express inequality 1.1 using the form of operator as follows: Hsu and Wang 33 also raised an open question how to obtain the best value θ of 8.7 .In 1992, Gao 34 gave the best value θ θ 0 1.281669 .Xu and Gao 35 proved a strengthened version of 2.6 given by Many diff118,119inds of Hilbert-type discrete and integral inequalities with applications are presented in this paper with references.Special attention is given to new results proved during 2009-2012.Included are many generalizations, extensions and refinements of Hilbert-type discrete and integral inequalities involving many special functions such as beta, gamma, hypergeometric, trigonometric, hyperbolic, zeta, Bernoulli's functions and Bernoulli's numbers, and Euler's constant.2Inhisthree books,Yang 116,118,119presented many new results on integral and discrete-type operators with general homogeneous kernels of degree of real numbers and two pairs of conjugate exponents as well as the related inequalities.These research monographs contained recent developments of both discrete and integral types of operators and inequalities with proofs, examples, and applications.