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

Abstract

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.

“As long as a branch of knowledge offers an abundance of problems, it is full of vitality”
David Hilbert

1. 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 the twentieth century through the pioneering work entitled Inequalities by G. H. Hardy, J. E. Littlewood, and G. Pòlya, 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=1𝑎²_𝑚<∞ and 0<∑^∞_𝑛=1𝑏²_𝑛<∞, then the Hilbert's double series inequality is given by ^∞∞_𝑛=1𝑚=1𝑎_𝑚𝑏_𝑛𝑚+𝑛<𝜋^∞_𝑚=1𝑎²_𝑚^∞_𝑛=1𝑏²_𝑛^1/2. (1.1) 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 ^∞₀^∞₀𝑓(𝑥)𝑔𝑦𝑥+𝑦𝑑𝑥𝑑𝑦<𝜋^∞₀𝑓²(𝑥)d𝑥^∞₀𝑔²𝑦𝑑𝑦^1/2, (1.2) where 𝑓 and 𝑔 are measurable functions such that 0<∫^∞₀𝑓²(𝑥)𝑑𝑥<∞ and 0<∫^∞₀𝑔²(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 _{ℝ𝑛ℝ𝑛}𝑓(𝑥)𝑔𝑦𝑥−𝑦^𝜆𝑑𝑥𝑑𝑦≤𝐶_{𝑛,𝜆,𝑝}𝑓_𝑝𝑔_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 𝐶_{𝑝,𝜆,𝑛}=𝐶_𝑛,𝜆=𝜋^𝜆/2Γ((𝑛−𝜆)/2)Γ((Γ()𝑛−𝜆/2))𝑛/2Γ(𝑛)^{(𝜆/𝑛)−1}, (1.4) and the equality in (1.3) holds if and only if 𝑔(𝑥)=𝑐₁𝑓(𝑥) and 𝑓(𝑥)=𝑐₂ℎ((𝑥/𝜇²)−𝑎), where ℎ(𝑥)=(1+|𝑥|²)^−𝑑, 2𝑑=(𝜆−2𝑛), 𝑎∈ℝ^𝑛, 𝑐₁, 𝑐₂, 𝜇∈ℝ/{0}.

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): _{ℝ𝑛ℝ𝑛}𝑓(𝑥)𝑔(𝑥)|𝑥|^𝛼𝑥−𝑦^𝜆𝑦^𝛽𝑑𝑥𝑑𝑦≤𝐶_{𝛼,𝛽,𝑛,𝜆,𝑝}𝑓_𝑝𝑔_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.

2. Operator Formulation of Hilbert's Inequality

Suppose that ℝ is the set of real numbers and ℝ^𝑚₊=(0,∞)×⋯^𝑚×(0,∞), for 𝑝>1,𝑙^𝑝∶={𝑎={𝑎_𝑛}^∞_𝑛=1|||𝑎||_𝑝={∑^∞_𝑛=1|𝑎_𝑛|^𝑝}^1/𝑝<∞} and 𝐿^𝑝(ℝ₊)∶={𝑓|||𝑓||_𝑝={∫^∞₀|𝑓(𝑥)|^𝑝𝑑𝑥}^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: 𝑇∶𝑙²→𝑙² is a linear operator, for any 𝑎={𝑎_𝑚}^∞_𝑚=1∈𝑙², there exists a sequence 𝑐={𝑐_𝑛}^∞_𝑛=1∈𝑙², satisfying 𝑐_𝑛=(𝑇𝑎)(𝑛)=^∞_𝑚=1𝑎_𝑚𝑚+𝑛,𝑛∈ℕ, (2.1) where ℕ is the set of positive integers. Hence, for any sequence 𝑏={𝑏_𝑛}^∞_𝑛=1∈𝑙², we define the inner product of 𝑇𝑎 and 𝑏 as follows: (𝑇𝑎,𝑏)=(𝑐,𝑏)=^∞_𝑛=1^∞_𝑚=1𝑎_𝑚𝑏𝑚+𝑛^𝑛=^∞∞_𝑛=1𝑚=1𝑎_𝑚𝑏_𝑛.𝑚+𝑛 (2.2) Using (2.2), inequality (1.1) can be rewritten in the operator form (𝑇𝑎,𝑏)<𝜋‖𝑎‖₂‖𝑏‖₂, (2.3) 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, ^∞_𝑛=1^∞_𝑚=1𝑎_𝑚𝑚+𝑛²<𝜋^2∞_𝑛=1𝑎²_𝑛, (2.4) where the constant factor 𝜋² 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: 𝑇∶𝐿²(ℝ₊)→𝐿²(ℝ₊), for any 𝑓∈𝐿²(ℝ₊), there exists a function, ℎ=𝑇𝑓∈𝐿²(ℝ₊), satisfying 𝑇𝑓𝑦=ℎ𝑦=^∞₀𝑓(𝑥)().𝑥+𝑦𝑑𝑥,𝑦∈0,∞ (2.5) Hence, for any 𝑔∈𝐿²(ℝ₊), we may still define the inner product of 𝑇𝑓 and 𝑔 as follows: 𝑇𝑓,𝑔=^∞₀^∞₀𝑓(𝑥)𝑔𝑦𝑥+𝑦𝑑𝑥𝑑𝑦=^∞₀^∞₀𝑓(𝑥)𝑔𝑦𝑥+𝑦𝑑𝑥𝑑𝑦. (2.6) Setting the norm of 𝑓 as ‖𝑓‖₂={∫^∞₀𝑓²(𝑥)𝑑𝑥}^1/2, if ‖𝑓‖₂ and ‖𝑔‖₂>0, then (1.2) may be rewritten in the operator form 𝑇𝑓,𝑔<𝜋𝑓₂𝑔₂. (2.7) It follows that ||𝑇𝑓||=𝜋 (see Carleman [11]), and we have the equivalent form of (1.2) as ‖𝑇𝑓‖₂<𝜋‖𝑓‖₂, (see Hardy et al. [3]), that is, ^∞₀^∞₀𝑓(𝑥)𝑥+𝑦𝑑𝑥²𝑑𝑦<𝜋^2∞₀𝑓²(𝑥)𝑑𝑥, (2.8) where the constant factor 𝜋² is still the best possible. It is obvious that inequality (2.8) is the integral analogue of (2.4).

3. 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: ^∞∞_𝑛=0𝑚=0𝑎_𝑚𝑏_𝑛𝑚+𝑛+2<𝜋^∞_𝑛=0𝑎²_𝑛^∞_𝑛=0𝑏²_𝑛^1/2, (3.1) 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]): ^∞∞_𝑛=0𝑚=0𝑎_𝑚𝑏_𝑛𝑚+𝑛+1<𝜋^∞_𝑛=0𝑎²_𝑛^∞_𝑛=0𝑏²_𝑛^1/2, (3.2) where the constant factor 𝜋 is the best possible.

Since for 𝑎_𝑚, 𝑏_𝑛≥0, 𝛼≥1, ^∞∞_𝑛=0𝑚=0𝑎_𝑚𝑏_𝑛≤𝑚+𝑛+𝛼^∞∞_𝑛=0𝑚=0𝑎_𝑚𝑏_𝑛,𝑚+𝑛+1 (3.3) then, by (3.2) and for 𝛼≥1, we obtain ^∞∞_𝑛=0𝑚=0𝑎_𝑚𝑏_𝑛𝑚+𝑛+𝛼<𝜋^∞_𝑛=0𝑎²_𝑛^∞_𝑛=0𝑏²_𝑛^1/2. (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: ^∞_𝑛=0^∞_𝑚=0𝑎_𝑚𝑚+𝑛+𝛼²<𝜋^2∞_𝑛=0𝑎²_𝑛,(1≤𝛼<2). (3.5) For 0<𝛼<1, in 1936, Ingham [12] proved that if 𝛼≥1/2, then ^∞∞_𝑛=0𝑚=0𝑎_𝑚𝑎_𝑛𝑚+𝑛+𝛼≤𝜋^∞_𝑛=0𝑎²_𝑛, (3.6) and if 0<𝛼<1/2, then ^∞∞_𝑛=0𝑚=0𝑎_𝑚𝑎_𝑛≤𝜋𝑚+𝑛+𝛼sin(𝛼𝜋)^∞_𝑛=0𝑎²_𝑛. (3.7)

Note 1. If we put 𝑥=𝑋+(𝛼/2), 𝑦=𝑌+(𝛼/2), 𝐹(𝑋)=𝑓(𝑋+(𝛼/2)), and 𝐺(𝑌)=𝑔(𝑌+(𝛼/2)) (𝛼∈ℝ) in (1.2), then we obtain ^∞_−𝛼/2^∞_−𝛼/2𝐹(𝑋)𝐺(𝑌)𝑋+𝑌+𝛼𝑑𝑋𝑑𝑌<𝜋^∞_−𝛼/2𝐹²(𝑋)𝑑𝑋^∞_−𝛼/2𝐺²(𝑌)𝑑𝑌^1/2. (3.8) For 𝛼≥1/2, inequality (3.8) is an integral analogue of (3.6) with 𝐺=𝐹. However, if 0<𝛼<1/2, inequality (3.8) is not an integral analogue of (3.7), because two constant factors are different.
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.

4. 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:

if 𝑝>1, 𝑎_𝑚, 𝑏_𝑛≥0, such that 0<∑^∞_𝑚=1𝑎^𝑝_𝑚<∞ and 0<∑^∞_𝑛=1𝑏^𝑞_𝑛<∞, then ^∞∞_𝑛=1𝑚=1𝑎_𝑚𝑏_𝑛<𝜋𝑚+𝑛sin𝜋/𝑝^∞_𝑚=1𝑎^𝑝_𝑚^1/𝑝∞_𝑛=1𝑏^𝑞_𝑛^1/𝑞, (4.1) where the constant factor 𝜋/sin(𝜋/𝑝) is the best possible. The equivalent discrete form of (4.1) is as follows: ^∞_𝑛=1^∞_𝑚=1𝑎_m𝑚+𝑛^𝑝<𝜋sin𝜋/𝑝^𝑝∞_𝑛=1𝑎^𝑝_𝑛, (4.2) 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]): ^∞∞_𝑛=0𝑚=0𝑎_𝑚𝑏_𝑛<𝜋𝑚+𝑛+1sin𝜋/𝑝^∞_𝑚=0𝑎^𝑝_𝑚^1/𝑝∞_𝑛=0𝑏^𝑞_𝑛^1/𝑞,_{∞𝑛=0∞𝑚=0}𝑎_𝑚𝑚+𝑛+1^𝑝<𝜋sin𝜋/𝑝^𝑝∞_𝑛=0𝑎^𝑝_𝑛, (4.3)(4.4) 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: ^∞₀^∞₀𝑓(𝑥)𝑔𝑦𝜋𝑥+𝑦𝑑𝑥𝑑𝑦<sin𝜋/𝑝^∞₀𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑔^𝑞𝑦𝑑𝑦^1/𝑞,_∞0∞0𝑓(𝑥)𝑥+𝑦𝑑𝑥^𝑝𝜋𝑑𝑦<sin𝜋/𝑝^𝑝∞₀𝑓^𝑝(𝑥)𝑑𝑥. (4.5)(4.6) We call (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 𝑇_𝑝𝑎=𝑐={𝑐_𝑛}^∞_𝑛=1∈𝑙^𝑝, satisfying 𝑐_𝑛=𝑇_𝑝𝑎(𝑛)=^∞_𝑚=0𝑎_𝑚𝑚+𝑛+1,𝑛∈ℕ₀=ℕ∪{0}. (4.7) And for any nonnegative sequence 𝑏={𝑏_𝑛}^∞_𝑛=1∈𝑙^𝑞, we can define the formal inner product of 𝑇_𝑝𝑎 and 𝑏 as follows: 𝑇_𝑝𝑎,𝑏=^∞_𝑛=0^∞_𝑚=0𝑎_𝑚𝑏𝑚+𝑛+1^𝑛=^∞∞_𝑛=0𝑚=0𝑎_𝑚𝑏_𝑛.𝑚+𝑛+1 (4.8) Then inequality (4.3) may be rewritten in the operator form 𝑇_𝑝<𝜋𝑎,𝑏‖𝑎‖sin𝜋/𝑝^𝑝‖𝑏‖_𝑞, (4.9) 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_𝑝𝑓∈𝐿^𝑝(ℝ₊), defined by 𝑇_𝑝𝑓𝑦=ℎ𝑦=^∞₀𝑓(𝑥)𝑥+𝑦𝑑𝑥,𝑦∈ℝ₊. (4.10) And for any 𝑔(≥0)∈𝐿^𝑞(ℝ₊), we can define the formal inner product of 𝑇_𝑝𝑓 and 𝑔 as follows: 𝑇_𝑝𝑓,𝑔=^∞₀^∞₀𝑓(𝑥)𝑔𝑦𝑥+𝑦𝑑𝑥𝑑𝑦. (4.11) Then inequality (4.5) may be rewritten as follows: 𝑇_𝑝<𝜋𝑓,𝑔𝑓sin𝜋/𝑝^𝑝𝑔_𝑞. (4.12)

On the other hand, if (𝑝,𝑞) is not a pair of conjugate exponents, then we have the following results (see Hardy et al. [3]).

If 𝑝>1,  𝑞>1, (1/𝑝)+(1/𝑞)≥1,  0<𝜆=2−((1/𝑝)+(1/𝑞))≤1, then ^∞∞_𝑛=1𝑚=1𝑎_𝑚𝑏_𝑛(𝑚+𝑛)^𝜆≤𝐾^∞_𝑚=1𝑎^𝑝_𝑚^1/𝑝∞_𝑛=1𝑏^𝑞_𝑛^1/𝑞, (4.13) 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 ^∞₀^∞₀𝑓(𝑥)𝑔𝑦𝑥+𝑦^𝜆𝑑𝑥𝑑𝑦≤𝐾^∞₀𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑔^𝑞𝑦𝑑𝑦^1/𝑞. (4.14) We also find an extension of (4.14) as follows (see Mitrinović et al. [4]):

If 𝑝>1, 𝑞>1, (1/𝑝)+(1/𝑞)>1, 0<𝜆=2−((1/𝑝)+(1/𝑞))<1, then ^∞_−∞^∞_−∞𝑓(𝑥)𝑔𝑦𝑥+𝑦^𝜆𝑑𝑥𝑑𝑦≤𝑘𝑝,𝑞^∞_−∞𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞_−∞𝑔^𝑞(𝑥)𝑑𝑥^1/𝑞. (4.15) 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.

5. A Hilbert-Type Inequality with the General Homogeneous Kernel of Degree −1

If 𝛼∈ℝ, the function 𝑘(𝑥,𝑦) is measurable in ℝ²₊, satisfying for any 𝑥, 𝑦, 𝑢>0, 𝑘(𝑢𝑥,𝑢𝑦)=𝑢^𝛼𝑘(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 that 𝑝>1, (1/𝑝)+(1/𝑞)=1, 𝑘₁(𝑥,𝑦)(≥0) is a homogeneous function of degree −1 in ℝ²₊. If 𝑓(𝑥),𝑔(𝑦)≥0, 𝑓∈𝐿^𝑝(ℝ₊), 𝑔∈𝐿^𝑞(ℝ₊),  𝑘=∫^∞₀𝑘₁(𝑢,1)𝑢^−1/𝑝𝑑𝑢 is finite, then we have 𝑘=∫^∞₀𝑘₁(1,𝑢)𝑢^−1/𝑞𝑑𝑢 and the following equivalent integral inequalities: ^∞₀^∞₀𝑘₁𝑥,𝑦𝑓(𝑥)𝑔𝑦𝑑𝑥𝑑𝑦≤𝑘^∞₀𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑔^𝑞𝑦𝑑𝑦^1/𝑞,_∞0∞0𝑘₁𝑥,𝑦𝑓(𝑥)𝑑𝑥^𝑝𝑑𝑦≤𝑘^𝑝∞₀𝑓^𝑝(𝑥)𝑑𝑥, (5.1)(5.2) where the constant factor 𝑘 is the best possible. Moreover, if 𝑎_𝑚,𝑏_𝑛≥0,𝑎={𝑎_𝑚}^∞_𝑚=1∈𝑙^𝑝,𝑏={𝑏_𝑛}^∞_𝑛=1∈𝑙^𝑞, both 𝑘₁(𝑢,1)𝑢^−1/𝑝 and 𝑘₁(1,𝑢)𝑢^−1/𝑞 are decreasing in ℝ₊, then we have the following equivalent discrete forms: ^∞∞_𝑛=1𝑚=1𝑘₁(𝑚,𝑛)𝑎_𝑚𝑏_𝑛≤𝑘^∞_𝑚=1𝑎^𝑝_𝑚^1/𝑝∞_𝑛=1𝑏^𝑞_𝑛^1/𝑞,_{∞𝑛=1∞𝑚=1}𝑘₁(𝑚,𝑛)𝑎_𝑚^𝑝≤𝑘^𝑝∞_𝑛=1𝑎^𝑝_𝑛. (5.3)(5.4) For 0<𝑝<1, if 𝑘=∫^∞₀𝑘₁(𝑢,1)𝑢^−1/𝑝𝑑𝑢 is finite, then we have the reverses of (5.1) and (5.2).

Note 2. We have not seen any proof of (5.1)–(5.4) and the reverse examples in [3].
We call 𝑘₁(𝑥,𝑦) the kernel of (5.1) and (5.2). If all the integrals and series in the right-hand side of inequalities (5.1)–(5.4) are positive, then we can obtain the following particular examples (see Hardy et al. [3]):(1)for 𝑘₁(𝑥,𝑦)=1/(𝑥+𝑦) in (5.1)–(5.4), they reduce to (4.5), (4.6), (4.1), and (4.2);(2)if 𝑘₁(𝑥,𝑦)=1/max{𝑥,𝑦} in (5.1)–(5.4), they reduce the following two pairs of equivalent forms: ^∞₀^∞₀𝑓(𝑥)𝑔𝑦max𝑥,𝑦𝑑𝑥𝑑𝑦<𝑝𝑞^∞₀𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑔^𝑞𝑦𝑑𝑦^1/𝑞,_∞0∞0𝑓(𝑥)max𝑥,𝑦𝑑𝑥^𝑝𝑑𝑦<𝑝𝑞^𝑝∞₀𝑓^𝑝(𝑥)𝑑𝑥;_{∞∞𝑛=1𝑚=1}𝑎_𝑚𝑏_𝑛max{𝑚,𝑛}<𝑝𝑞^∞_𝑚=1𝑎^𝑝_𝑚^1/𝑝∞_𝑛=1𝑏^𝑞_𝑛^1/𝑞,_{∞𝑛=1∞𝑚=1}𝑎_𝑚max{𝑚,𝑛}^𝑝<𝑝𝑞^𝑝∞_𝑛=1𝑎^𝑝_𝑛; (5.5)(5.6)(5.7)(5.8)(3)if 𝑘₁(𝑥,𝑦)=ln(𝑥/𝑦)/(𝑥−𝑦) in (5.1)–(5.4), they reduce to the following two pairs of equivalent inequalities: ln𝑥/𝑦𝑓^∞₀^∞₀(𝑥)𝑔𝑦<𝜋𝑥−𝑦𝑑𝑥𝑑𝑦sin𝜋/𝑝^2∞₀𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑔^𝑞𝑦𝑑𝑦^1/𝑞,_∞0∞0ln𝑥/𝑦𝑓(𝑥)𝑥−𝑦𝑑𝑥^𝑝𝜋𝑑𝑦<sin𝜋/𝑝^2𝑝∞₀𝑓^𝑝(𝑥)𝑑𝑥;_{∞∞𝑛=1𝑚=1}ln(𝑚/𝑛)𝑎_𝑚𝑏_𝑛<𝜋𝑚−𝑛sin𝜋/𝑝^2∞_𝑚=1𝑎^𝑝_m^1/𝑝∞_𝑛=1𝑏^𝑞_𝑛^1/𝑞,_{∞𝑛=1∞𝑚=1}ln(𝑚/𝑛)𝑎_𝑚𝑚−𝑛^𝑝<𝜋sin𝜋/𝑝^∞2𝑝_𝑛=1𝑎^𝑝_𝑛. (5.9)(5.10)(5.11)(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]): ln𝑥/𝑦𝑓^∞₀^∞₀(𝑥)𝑔𝑦<𝑝max𝑥,𝑦𝑑𝑥𝑑𝑦₂+𝑞^2∞₀𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑔^𝑞𝑦𝑑𝑦^1/𝑞,_∞0∞0ln𝑥/𝑦𝑓(𝑥)max𝑥,𝑦𝑑𝑥^𝑝𝑑𝑦<𝑝²+𝑞^2𝑝∞₀𝑓^𝑝(𝑥)𝑑𝑥,_{∞∞𝑛=1𝑚=1}|ln(𝑚/𝑛)|𝑎_𝑚𝑏_𝑛{}<𝑝max𝑚,𝑛²+𝑞^2∞_𝑚=1𝑎^𝑝_𝑚^1/𝑝∞_𝑛=1𝑏^𝑞_𝑛^1/𝑞,_{∞𝑛=1∞𝑚=1}|ln(𝑚/𝑛)|𝑎_𝑚max{𝑚,𝑛}^𝑝<𝑝²+𝑞^2𝑝∞_𝑛=1𝑎^𝑝_𝑛, (5.13)(5.14)(5.15)(5.16) where the constant factors (𝑝²+𝑞²) and (𝑝²+𝑞²)^𝑝 are the best possible. Another type inequalities with the best constant factors are as follows (see Xin and Yang [20]): ln𝑥/𝑦𝑓^∞₀^∞₀(𝑥)𝑔𝑦𝑥+𝑦𝑑𝑥𝑑𝑦<𝑐₀𝑝^∞₀𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑔^𝑝𝑦𝑑𝑦^1/𝑞,_∞0∞0ln𝑥/𝑦𝑓(𝑥)𝑥+𝑦𝑑𝑥^𝑝𝑑𝑦<𝑐^𝑝₀𝑝^∞₀𝑓^𝑝(𝑥)𝑑𝑥;_{∞∞𝑛=1𝑚=1}|ln(𝑚/𝑛)|𝑎_𝑚𝑏_𝑛𝑚+𝑛<𝑐₀(2)^∞_𝑚=1𝑎²_𝑚^∞_𝑛=1𝑏²_𝑛^1/2,_{∞𝑛=1∞𝑚=1}|ln(𝑚/𝑛)|𝑎_𝑚𝑚+𝑛²<𝑐²₀(2)^∞_𝑛=1𝑎²_𝑛, (5.17)(5.18)(5.19)(5.20) where the constant factor 𝑐₀(𝑝)=2∑^∞_𝑛=1(−1)^𝑛−1[1/(𝑛−(1/𝑝))²−(1/(𝑛−(1/𝑞))²)].

6. 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, 𝑝⁻¹+𝑞⁻¹+⋯+𝑟⁻¹=1, and 𝑘(𝑥,𝑦,…,𝑧)≥0 is a homogeneous function of degree (−𝑛+1). If 𝑘=^∞₀^∞₀⋯^∞₀𝑘1,𝑦,…,𝑧𝑦^−1/𝑞⋯𝑧^−1/𝑟𝑑𝑦⋯𝑑𝑧 (6.1) is a finite number, 𝑓,𝑔,…,ℎ are nonnegative measurable functions in ℝ₊, then, we have the following multiple Hilbert-type integral inequality (see Hardy et al. [3]): ^∞₀^∞₀⋯^∞₀𝑘𝑥,𝑦,…,𝑧𝑓(𝑥)𝑔𝑦⋯ℎ(𝑧)𝑑𝑥𝑑𝑦⋯𝑑𝑧≤𝑘^∞₀𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑔^𝑞𝑦𝑑𝑦^1/𝑞⋯^∞₀ℎ^𝑟(𝑧)𝑑𝑧^1/𝑟. (6.2) Moreover, if 𝑎_𝑚, 𝑏_𝑛,…,𝑐_𝑠≥0, 𝑘(1,𝑦,…,𝑧)𝑥⁰𝑦^−1/𝑞⋯𝑧^−1/𝑟,𝑘(𝑥,1,…,𝑧)𝑥^−1/𝑝𝑦⁰⋯𝑧^−1/𝑟,…,𝑘(𝑥,𝑦,…,1)𝑥^−1/𝑝𝑦^−1/𝑞⋯𝑧⁰ are all decreasing functions with respect to any single variable in ℝ₊, then, we have ^∞_𝑠=1⋯^∞∞_𝑛=1𝑚=1𝑘(𝑚,𝑛,…,𝑠)𝑎_𝑚𝑏_𝑛⋯𝑐_𝑠≤𝑘^∞_𝑚=1𝑎^𝑝_𝑚^1/𝑝∞_𝑛=1𝑏^𝑞_𝑛^1/𝑞⋯^∞_𝑠=1𝑐^𝑟_𝑠^1/𝑟. (6.3)

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).

7. Modern Research for Hilbert's Integral Inequality

In 1979, based on an improvement of Hölder's inequality, Hu [21] proved a refinement of (1.2) (for 𝑓=𝑔) as follows: ^∞₀^∞₀𝑓(𝑥)𝑓𝑦𝑥+𝑦𝑑𝑥𝑑𝑦<𝜋^∞₀𝑓²(𝑥)𝑑𝑥²−14^∞₀𝑓²(𝑥)cos√𝑥𝑑𝑥^21/2. (7.1) 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, ^𝑎₀^𝑏₀𝑓(𝑥)𝑔𝑦√𝑥+𝑦𝑑𝑥𝑑𝑦<𝑎𝑏2^𝑎₀(𝑎−𝑥)𝑓^′2(𝑥)𝑑𝑥^𝑏₀(𝑏−𝑥)𝑔^′2𝑦𝑑𝑦^1/2. (7.2) 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: ^𝑏_𝑎^𝑏_𝑎𝑓(𝑥)𝑔𝑦𝑥+𝑦^𝜆𝜆𝑑𝑥𝑑𝑦<𝐵2,𝜆2𝑎1−𝑏^𝜆/4𝑏_𝑎𝑥^1−𝜆𝑓²(𝑥)𝑑𝑥^𝑏_𝑎𝑥^1−𝜆𝑔²𝑦𝑑𝑦^1/2, (7.3) where 𝐵(𝑢,𝑣) is the beta function.

In 1999, Kuang [28] gave another extension of (1.2) as follows: for 𝜆∈(1/2,1], ^∞₀^∞₀𝑓(𝑥)𝑓𝑦𝑥^𝜆+𝑦^𝜆𝜋𝑑𝑥𝑑𝑦<𝜆sin(𝜋/2𝜆)^∞₀𝑓²(𝑥)𝑑𝑥^∞₀𝑔²𝑦𝑑𝑦^1/2. (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: ^∞₀^∞₀𝑓(𝑥)𝑓𝑦√𝑥+𝑦𝑑𝑥𝑑𝑦<𝜋1−𝑅^∞₀𝑓²(𝑥)𝑑𝑥^∞₀𝑔²𝑦𝑑𝑦^1/2, (7.5) where ||𝑓||=∫^∞₀𝑓²(𝑥)𝑑𝑥^1/2, 𝑅=1/𝜋((𝑢/‖𝑔‖)−(𝑣/‖𝑓‖))², 𝑢=(2/𝜋)(𝑔,𝑒), 𝑣=√2𝜋(𝑓,𝑒^−𝑥),  𝑒(𝑦)=∫^∞₀(𝑒^𝑥/(𝑥+𝑦))𝑑𝑥. We also refer to works of Gao and Hsu in [31].

In 2002, using the operator theory, Zhang [32] gave an improvement of (1.2) as follows: ^∞₀^∞₀𝑓(𝑥)𝑓𝑦≤𝜋𝑥+𝑦𝑑𝑥𝑑𝑦√2^∞0𝑓²(𝑥)𝑑𝑥^∞₀𝑔²(𝑥)𝑑𝑥+^∞₀𝑓(𝑥)𝑔(𝑥)𝑑𝑥^21/2. (7.6)

8. 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 𝐼=^∞∞_𝑛=1𝑚=1𝑎_𝑚𝑏_𝑛=𝑚+𝑛^∞∞_𝑛=1𝑚=11𝑚𝑚+𝑛𝑛^1/4𝑎_𝑚𝑛𝑚^1/4𝑏_𝑛≤^∞_𝑚=1^∞_𝑛=11𝑚𝑚+𝑛𝑛^1/2𝑎²_𝑚^∞_𝑛=1^∞_𝑚=11𝑚𝑚+𝑛𝑛^1/2𝑏²_𝑛^1/2. (8.1) Then, we define the weight coefficient 𝜔(𝑛)=^∞_𝑚=11𝑚𝑚+𝑛𝑛^1/2,𝑛∈ℕ, (8.2) and rewrite (8.1) as follows: 𝐼≤^∞_𝑚=1𝜔(𝑚)𝑎²_𝑚^∞_𝑛=1𝜔(𝑛)𝑏²_𝑛^1/2. (8.3) Setting 𝜔(𝑛)𝜃(𝑛)=𝜋−𝑛^1/2,𝑛∈ℕ, (8.4) where 𝜃(𝑛)=(𝜋−𝜔(𝑛))𝑛^1/2, and estimating the series of 𝜃(𝑛), it follows that 𝜃(𝑛)=𝜋−^∞_𝑚=11𝑚𝑚+𝑛𝑛^1/2𝑛^1/2>𝜃=1.1213⁺. (8.5) Thus, result (8.4) yields 𝜔(𝑛)𝜃<𝜋−𝑛^1/2,𝑛∈ℕ,𝜃=1.1213⁺. (8.6) In view of (8.3), a strengthened version of (1.1) is given by 𝐼<^∞_𝑛=1𝜃𝜋−𝑛^1/2𝑎²_𝑛^∞_𝑛=1𝜃𝜋−𝑛^1/2𝑏²_𝑛^1/2. (8.7) 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=𝜃₀=1.281669⁺.

Xu and Gao [35] proved a strengthened version of (2.6) given by 𝐼<^∞_𝑛=1𝜋−sin𝜋/𝑝𝑝−1𝑛^1/𝑝+𝑛^−1/𝑞𝑎^𝑝_𝑛^1/𝑝×^∞_𝑛=1𝜋−sin𝜋/𝑝𝑞−1𝑛^1/𝑞+𝑛^−1/𝑝𝑏^𝑞_𝑛^1/𝑞. (8.8) In 1997, using the way of weight coefficient and the improved Euler-Maclaurin's summation formula, Yang and Gao [36, 37] showed that 𝐼<^∞_𝑛=1𝜋−sin𝜋/𝑝1−𝛾𝑛^1/𝑝𝑎^𝑝_𝑛^1/𝑝∞_𝑛=1𝜋−sin𝜋/𝑝1−𝛾𝑛^1/𝑞𝑏^𝑞_𝑛^1/𝑞, (8.9) 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].

9. 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<∫^∞₀𝑥^1−𝜆𝑓²(𝑥)𝑑𝑥<∞ and 0<∫^∞₀𝑥^1−𝜆𝑔²(𝑥)𝑑𝑥<∞, then ^∞₀^∞₀𝑓(𝑥)𝑔𝑦𝑥+𝑦^𝜆𝜆𝑑𝑥𝑑𝑦<𝐵2,𝜆2^∞₀𝑥^1−𝜆𝑓²(𝑥)𝑑𝑥^∞₀𝑥^1−𝜆𝑔²(𝑥)𝑑𝑥^1/2, (9.1) 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]: 𝐵(𝑢,𝑣)=^∞₀𝑡^𝑢−1𝑑𝑡(1+𝑡)^𝑢+𝑣=¹₀(1−𝑡)^𝑢−1𝑡^𝑣−1=𝑑𝑡_∞1(𝑡−1)^𝑢−1𝑑𝑡𝑡^𝑢+𝑣,(𝑢,𝑣>0). (9.2) Some extensions of (4.1), (4.3), and (4.5) were given by Yang and Debnath [43–45] as follows. If 𝜆>2−min{𝑝,𝑞}, then ^∞₀^∞₀𝑓(𝑥)𝑔𝑦𝑥+𝑦^𝜆𝑑𝑥𝑑𝑦<𝐵𝑝+𝜆−2𝑝,𝑞+𝜆−2𝑞×^∞₀𝑥^1−𝜆𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑥^1−𝜆𝑔^𝑞(𝑥)𝑑𝑥^1/𝑞. (9.3) If 2−min{𝑝,𝑞}<𝜆≤2, then ^∞∞_𝑛=1𝑚=1𝑎_𝑚𝑏_𝑛(𝑚+𝑛)^𝜆<𝐵𝑝+𝜆−2𝑝,𝑞+𝜆−2𝑞×^∞_𝑛=1𝑛^1−𝜆𝑎^𝑝_𝑛^1/𝑝∞_𝑛=1𝑛^1−𝜆𝑏^𝑞_𝑛^1/𝑞,_{∞∞𝑛=0𝑚=0}𝑎_𝑚𝑏_𝑛(𝑚+𝑛+1)^𝜆<𝐵𝑝+𝜆−2𝑝,𝑞+𝜆−2𝑞×^∞_𝑛=01𝑛+2^1−𝜆𝑎^𝑝_𝑛^1/𝑝∞_𝑛=01𝑛+2^1−𝜆𝑏^𝑞_𝑛^1/𝑞, (9.4)(9.5) where the constant factor 𝐵((𝑝+𝜆−2)/𝑝,(𝑞+𝜆−2)/𝑞) is the best possible. Yang [46] also proved that (9.4) is valid for 𝑝=2 and 𝜆∈(0,4]. Yang [47, 48] gave another extensions of (4.1) and (4.3) as follows: if 0<𝜆≤min{𝑝,𝑞}, then ^∞∞_𝑛=1𝑚=1𝑎_𝑚𝑏_𝑛𝑚^𝜆+𝑛^𝜆<𝜋𝜆sin𝜋/𝑝^∞_𝑛=1𝑛^{(𝑝−1)(1−𝜆)}𝑎^𝑝_𝑛^1/𝑝∞_𝑛=1𝑛^{(𝑞−1)(1−𝜆)}𝑏^𝑞_𝑛^1/𝑞 (9.6) and if 0<𝜆≤1, then ^∞∞_𝑛=0𝑚=0𝑎_𝑚𝑏_𝑛(𝑚+(1/2))^𝜆+(𝑛+(1/2))^𝜆<𝜋𝜆sin𝜋/𝑝^∞_𝑛=0(𝑛+(1/2))^{𝑝−1−𝜆}𝑎^𝑝_𝑛^1/𝑝∞_𝑛=0(𝑛+(1/2))^{𝑞−1−𝜆}𝑏^𝑞_𝑛^1/𝑞. (9.7) In 2004, Yang [49] proved the following dual form of (4.1): ^∞∞_𝑛=1𝑚=1𝑎_𝑚𝑏_𝑛<𝜋𝑚+𝑛sin𝜋/𝑝^∞_𝑛=1𝑛^𝑝−2𝑎^𝑝_𝑛^1/𝑝∞_𝑛=1𝑛^𝑞−2𝑏^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: ^∞∞_𝑛=0𝑚=0𝑎_𝑚𝑏_𝑛<𝜋𝑚+𝑛+1sin𝜋/𝑝^∞_𝑛=0(𝑛+(1/2))^𝑝−2𝑎^𝑝_𝑛^1/𝑝∞_𝑛=0(𝑛+(1/2))^𝑞−2𝑏^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 (4.1): if 𝑛∈𝐍⧵{1}, 𝑝_𝑖>1, ∑^𝑛_𝑖=11/𝑝_𝑖=1, 𝜆>𝑛−min_{1≤𝑖≤𝑛}{𝑝_𝑖}, 𝑓_𝑖(𝑡)≥0 and 0<∫^∞₀𝑡^{𝑛−1−𝜆}𝑓^𝑝𝑖_𝑖(𝑡)𝑑𝑡<∞(𝑖=1,2,…,𝑛), then we have ^∞₀⋯^∞₀∏^𝑛_𝑖=1𝑓_𝑖𝑥_𝑖∑^𝑛_𝑖=1𝑥_𝑖^𝜆𝑑𝑥₁⋯𝑑𝑥_𝑛<1Γ(𝜆)^𝑛_𝑖=1𝑝_𝑖+𝜆−𝑛𝑝_𝑖^∞₀𝑡^{𝑛−1−𝜆}𝑓^𝑝𝑖_𝑖(𝑡)𝑑𝑡^1/𝑝𝑖, (9.10) where the constant factor 1/Γ(𝜆)∏^𝑛_𝑖=1((𝑝_𝑖+𝜆−𝑛)/𝑝_𝑖) is the best possible. In particular, for 𝜆=𝑛−1, it follows that ^∞₀⋯^∞₀∏^𝑛_𝑖=1𝑓_𝑖𝑥_𝑖∑^𝑛_𝑖=1𝑥_𝑖^𝑛−1𝑑𝑥₁⋯𝑑𝑥_𝑛<1Γ(𝜆)^𝑛_𝑖=111−𝑝_𝑖^∞₀𝑓^𝑝𝑖_𝑖(𝑡)𝑑𝑡^1/𝑝𝑖. (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.

10. 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 ^∞₀^∞₀𝑓(𝑥)𝑔𝑦𝑥^𝜆+𝑦^𝜆<𝜋𝑑𝑥𝑑𝑦𝜆sin(𝜋/𝑟)^∞₀𝑥^{𝑝(1−(𝜆/𝑟))−1}𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑥^{𝑞(1−(𝜆/𝑠))−1}𝑔^𝑞(𝑥)𝑑𝑥^1/𝑞, (10.1) where the constant factor 𝜋/𝜆sin(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: ^∞₀^∞₀𝑓(𝑥)𝑔𝑦<𝜋𝑥+𝑦𝑑𝑥𝑑𝑦sin𝜋/𝑝^∞₀𝑥^𝑝−2𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑥^𝑞−2𝑔^𝑞(𝑥)𝑑𝑥^1/𝑞. (10.2)

In 2005, by introducing an independent parameter 𝜆>0, and two pairs of generalized conjugate exponents (𝑝₁,𝑝₂,…,𝑝_𝑛) and (𝑟₁,𝑟₂,…,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:

for 𝑝_𝑖,𝑟_𝑖>1(𝑖=1,2,…,𝑛), ^∞₀⋯^∞₀∏^𝑛_𝑖=1𝑓_i𝑥_𝑖∑^𝑛_𝑖=1𝑥_𝑖^𝜆𝑑𝑥₁⋯𝑑𝑥_𝑛<1Γ(𝜆)^𝑛_𝑖=1𝜆𝑟_𝑖^∞₀𝑡^{𝑝𝑖(1−(𝜆/𝑟𝑖))−1}𝑓^𝑝𝑖_𝑖(𝑡)𝑑𝑡^1/𝑝𝑖, (10.3) where the constant factor 1/Γ(𝜆)∏^𝑛_𝑖=1(𝜆/𝑟_𝑖) is the best possible. For 𝑛=2, 𝑝₁=𝑝, 𝑝₂=𝑞, 𝑟₁=𝑟, and 𝑟₂=𝑠, inequality (10.3) reduces to the following: ^∞₀^∞₀𝑓(𝑥)𝑔𝑦𝑥+𝑦^𝜆𝜆𝑑𝑥𝑑𝑦<𝐵𝑟,𝜆𝑠^∞₀𝑥^{𝑝(1−(𝜆/𝑟))−1}𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑥^{𝑞(1−(𝜆/𝑠))−1}𝑔^𝑞(𝑥)𝑑𝑥^1/𝑞. (10.4) It is obvious that inequality (10.4) is another best extension of (4.5).

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 ℝ^𝑛₊={𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛);𝑥_𝑖>0,𝑖=1,2,…,𝑛}, 𝛼, 𝛽, 𝜆>0, ‖𝑥‖_𝛼=(∑^𝑛_𝑖=1𝑥^𝛼_𝑖)^1/𝛼, 𝑓, 𝑔≥0, 0<∫_ℝ𝑛+‖𝑥‖_𝛼^{𝑝(𝑛−(𝛽𝜆/𝑟))−𝑛}𝑓^𝑝(𝑥)𝑑𝑥<∞ and 0<∫_ℝ𝑛+‖𝑥‖_𝛼^{𝑞(𝑛−(𝛽𝜆/𝑠))−𝑛}𝑔^𝑞(𝑥)𝑑𝑥<∞, then _{ℝ𝑛+ℝ𝑛+}𝑓(𝑥)𝑔𝑦𝑑𝑥𝑑𝑦‖𝑥‖^𝛽_𝛼+𝑦^𝛽_𝛼^𝜆<Γ^𝑛(1/𝛼)𝛽𝛼^𝑛−1Γ()𝐵𝜆𝑛/𝛼𝑟,𝜆𝑠_ℝ^𝑛₊‖𝑥‖_𝛼^{𝑝(𝑛−(𝛽𝜆/𝑟))−𝑛}𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝×_ℝ𝑛+‖𝑥‖_𝛼^{𝑞(𝑛−(𝛽𝜆/𝑠))−𝑛}𝑔^𝑞(𝑥)𝑑𝑥^1/𝑞, (10.5) where the constant factor Γ^𝑛(1/𝛼)/(𝛽𝛼^𝑛−1Γ(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]): ^∞∞_𝑛=1𝑚=1𝑎_𝑚𝑏_𝑛(𝑚^𝛼+𝑛^𝛼)^𝜆<1𝛼𝐵𝜆𝑟,𝜆𝑠^∞_𝑛=1𝑛^{𝑝(1−(𝛼𝜆/𝑟))−1}𝑎^𝑝_𝑛^1/𝑝∞_𝑛=1𝑛^{𝑞(1−(𝛼𝜆/𝑠))−1}𝑏^𝑞_𝑛^1/𝑞, (10.6) where 𝛼, 𝜆>0, 𝛼𝜆≤min{𝑟,𝑠}. In particular, for 𝛼=1, we have ^∞∞_𝑛=1𝑚=1𝑎_𝑚𝑏_𝑛(𝑚+𝑛)^𝜆𝜆<𝐵𝑟,𝜆𝑠^∞_𝑛=1𝑛^{𝑝(1−(𝜆/𝑟))−1}𝑎^𝑝_𝑛^1/𝑝∞_𝑛=1𝑛^{𝑞(1−(𝜆/𝑠))−1}𝑏^𝑞_𝑛^1/𝑞. (10.7) 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]): ^∞∞_𝑛=0𝑚=0𝑎_𝑚𝑏_𝑛(𝑚+𝑛+1)²>2^∞_𝑛=011−4(𝑛+1)²𝑎^𝑝_𝑛2𝑛+1^1/𝑝∞_𝑛=0𝑏^𝑞_𝑛2𝑛+1^1/𝑞, (10.8) 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: ln𝑥/𝑦𝑓^∞₀^∞₀(𝑥)𝑔𝑦𝑥^𝜆−𝑦^𝜆𝜋𝑑𝑥𝑑𝑦<sin(𝜋/𝑟)^2∞₀𝑥^{𝑝(1−(𝜆/𝑟))−1}𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝×^∞₀𝑥^{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: ^∞₀^∞₀𝑓(𝑥)𝑔𝑦max𝑥^𝜆,𝑦^𝜆<𝑑𝑥𝑑𝑦𝑟𝑠𝜆^∞₀𝑥^{𝑝(1−(𝜆/𝑟))−1}𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑥^{𝑞(1−(𝜆/𝑠))−1}𝑔^𝑞(𝑥)𝑑𝑥^1/𝑞. (10.10) Zhong and Yang [89] also gave the reverse form of (10.10).

Considering a particular kernel, Yang [90] proved ^∞∞_𝑛=1𝑚=1𝑎_𝑚𝑏_𝑛√𝑚+√𝑛max{𝑚,𝑛}<4ln2^∞_𝑛=1𝑛^(𝑝/2)−1𝑎^𝑝_𝑛^1/𝑝∞_𝑛=1𝑛^(𝑞/2)−1𝑏^𝑞_𝑛^1/𝑞. (10.11) Yang [91] also proved that ^∞∞_𝑛=1𝑚=1𝑎_𝑚𝑏_𝑛(𝑚+𝑎𝑛)²+𝑛²<𝜋2−arctan𝑎^∞_𝑛=1𝑎^𝑝_𝑛𝑛^1/𝑝∞_𝑛=1𝑏^𝑞_𝑛𝑛^1/𝑞,(𝑎≥0). (10.12) Using the residue theory, Yang [92] obtained the following inequality: ^∞₀^∞₀𝑓(𝑥)𝑔𝑦𝑥+𝑎𝑦𝑥+𝑏𝑦𝑥+𝑐𝑦𝑑𝑥𝑑𝑦<𝑘^∞₀𝑥^{(−𝑝/2)−1}𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑥^{(−𝑞/2)−1}𝑔^𝑞(𝑥)𝑑𝑥^1/𝑞, (10.13) where 𝑘=1/((√𝑎+√𝑏)(√𝑏+√𝑐)(√𝑎+√𝑐))(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 (see [75, 93–97]).

11. 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: (𝑎,𝑇𝑏)²≤‖𝑇‖²2‖𝑎‖²‖𝑏‖²+(𝑎,𝑏)²,(𝑎,𝑏∈𝐻), (11.1) 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: ^∞∞_𝑛=0𝑚=0𝑎_𝑚𝑏_𝑛<𝜋𝑚+𝑛+1√2^∞_𝑛=0𝑎²_𝑛^∞_𝑛=0𝑏²_𝑛+^∞_𝑛=0𝑎_𝑛𝑏_𝑛^21/2. (11.2)

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 ℝ²₊ satisfying ^∞₀𝑘()𝑥𝑥,𝑡𝑡^1/𝑟𝑑𝑡=𝑘₀𝑝∈ℝ,(𝑥>0). (11.3) We define an integral operator as 𝑇∶𝐿^𝑟ℝ₊→𝐿^𝑟ℝ₊,𝑟=𝑝,𝑞, (11.4) for any 𝑓(≥0)∈𝐿^𝑝(ℝ₊), there exists ℎ=𝑇𝑓∈𝐿^𝑝(ℝ₊), such that 𝑇𝑓𝑦=ℎ𝑦∶=^∞₀𝑘𝑥,𝑦𝑓(𝑥)𝑑𝑥,𝑦>0. (11.5) Or, for any 𝑔(≥0)∈𝐿^𝑞(ℝ₊), there exists ℎ=𝑇𝑔∈𝐿^𝑞(ℝ₊), such that 𝑇𝑔(𝑥)=ℎ(𝑥)∶=^∞₀𝑘𝑥,𝑦𝑔𝑦𝑑𝑦,(𝑥>0). (11.6)

In 2006, Yang [98] proved that the operator 𝑇 defined by (11.5) or (11.6) are bounded with ‖𝑇‖≤𝑘₀(𝑝). The following are some results in this paper: If 𝜀>0 is small enough and the integral ∫^∞₀𝑘(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)=𝑘₀(𝑝)+𝑜(1)(𝜀→0⁺), then ‖𝑇‖=𝑘₀(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_𝑝𝑔_𝑞,𝑇𝑓_𝑝<‖𝑇‖⋅𝑓_𝑝. (11.7)(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‖=𝑘₀(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]).

12. 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 ^∞₀^∞₀𝑓(𝑥)𝑔𝑦𝑥+𝑦𝑑𝑥𝑑𝑦<𝜋^∞₀𝑓²(𝑥)𝑑𝑥^∞₀𝑔²(𝑥)𝑑𝑥^1/2, (12.1) and the following H-L-P inequalities (for 𝑝=2 in (5.5) and (5.10)): ^∞₀^∞₀𝑓(𝑥)𝑔𝑦max𝑥,𝑦𝑑𝑥𝑑𝑦<4^∞₀𝑓²(𝑥)𝑑𝑥^∞₀𝑔²(𝑥)𝑑𝑥^1/2,_∞0∞0ln𝑥/𝑦𝑓(𝑥)𝑔𝑦𝑥−𝑦𝑑𝑥𝑑𝑦<𝜋^2∞₀𝑓²(𝑥)𝑑𝑥^∞₀𝑔²(𝑥)𝑑𝑥^1/2 (12.2)(12.3) basic Hilbert-type integral inequalities. In 2006, Yang [115] gave the following basic Hilbert-type integral inequality: ln𝑥/𝑦𝑓^∞₀^∞₀(𝑥)𝑔𝑦max𝑥,𝑦𝑑𝑥𝑑𝑦<8^∞₀𝑓²(𝑥)𝑑𝑥^∞₀𝑔²(𝑥)𝑑𝑥^1/2. (12.4) In 2011, Yang [116] gave the following basic Hilbert-type integral inequalities: ln𝑥/𝑦𝑓^∞₀^∞₀(𝑥)𝑔𝑦𝑥+𝑦𝑑𝑥𝑑𝑦<𝑐₀^∞₀𝑓²(𝑥)𝑑𝑥^∞₀𝑔²(𝑥)𝑑𝑥^1/2,_∞0∞0arctan𝑥/𝑦𝑓(𝑥)𝑔𝑦𝜋𝑥+𝑦𝑑𝑥𝑑𝑦<²4^∞₀𝑓²(𝑥)𝑑𝑥^∞₀𝑔²(𝑥)𝑑𝑥^1/2, (12.5)(12.6) where 𝑐₀=8∑^∞_𝑛=1(−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): ^∞₀^∞₀𝑓(𝑥)𝑔𝑦𝑥−𝑦^𝜆𝜆𝑑𝑥𝑑𝑦<2𝐵1−𝜆,2^∞₀𝑥^1−𝜆𝑓²(𝑥)𝑑𝑥^∞₀𝑥^1−𝜆𝑔²(𝑥)𝑑𝑥^1/2. (12.7) Similar to discrete inequality (3.7), the following inequality: ^∞₀^∞₀𝑓(𝑥)𝑔𝑦𝑥+𝑦^𝜆𝜆𝑑𝑥𝑑𝑦<𝐵2,𝜆2^∞₀𝑥^1−𝜆𝑓²(𝑥)𝑑𝑥^∞₀𝑥^1−𝜆𝑔²(𝑥)𝑑𝑥^1/2 (12.8) is called basic Hilbert-type integral inequality with a parameter 𝜆∈(0,∞).

Also we find the following basic Hilbert-type inequalities: ^∞∞_𝑛=1𝑚=1𝑎_𝑚𝑏_𝑛𝑚+𝑛<𝜋^∞_𝑛=1𝑎²_𝑛^∞_𝑛=1𝑏²_𝑛^1/2,_{∞∞𝑛=1𝑚=1}𝑎_𝑚𝑏_𝑛max{𝑚,𝑛}<4^∞_𝑛=1𝑎²_𝑛^∞_𝑛=1𝑏²_𝑛^1/2,_{∞∞𝑛=1𝑚=1}ln(𝑚/𝑛)𝑎_𝑚𝑏_𝑛𝑚−𝑛<𝜋^2∞_𝑛=1𝑎²_𝑛^∞_𝑛=1𝑏²_𝑛^1/2. (12.9)(12.10)(12.11) It follows from (5.15) with 𝑝=𝑞=2 that ^∞∞_𝑛=1𝑚=1|ln(𝑚/𝑛)|𝑎_𝑚𝑏_𝑛max{𝑚,𝑛}<8^∞_𝑛=1𝑎²_𝑛^∞_𝑛=1𝑏²_𝑛^1/2. (12.12) In 2010, Xin and Yang [20] proved the following inequality: ^∞∞_𝑛=1𝑚=1|ln(𝑚/𝑛)|𝑎_𝑚𝑏_𝑛𝑚+𝑛<𝑐₀^∞_𝑛=1𝑎²_𝑛^∞_𝑛=1𝑏²_𝑛^1/2, (12.13) where 𝑐₀=8∑^∞_𝑛=1(−1)^𝑛/(2𝑛−1)²=7.3277⁺. Inequalities (12.12) and (12.13) are new basic Hilbert-type inequalities. We still have a basic Hilbert-type inequality with a parameter 𝜆∈(0,4] as follows (see Yang [47]): ^∞∞_𝑛=1𝑚=1𝑎_𝑚𝑏_𝑛(𝑚+𝑛)^𝜆𝜆<𝐵2,𝜆2^∞_𝑛=1𝑛^1−𝜆𝑎²_𝑛^∞_𝑛=1𝑛^1−𝜆𝑏²_𝑛^1/2. (12.14)

13. Some Recent Work with Applications

In 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 Hilbert-type 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 Hilbert-type 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(𝜆₁,𝜆₂): ^∞₀𝑓(𝑥)^∞_𝑛=1𝑎_𝑛(𝑥+𝑛)^𝜆𝑑𝑥<𝐵𝜆₁,𝜆₂𝑓_𝑝,𝜙‖𝑎‖_𝑞,𝜓, (13.1) where 𝜆₁𝜆₂>0, 0≤𝜆₂≤1, 𝜆₁+𝜆₂=𝜆, 𝑓_𝑝,𝜙=^∞₀𝜑(𝑥)𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝‖𝑎‖>0,_𝑞,𝜓=^∞_𝑛=1𝑎^𝑞_𝑛^1/𝑞>0,𝜑(𝑥)=𝑥^{𝑝(1−𝜆1)−1}, (13.2)𝜓(𝑛)=𝑛^{𝑞(1−𝜆2)−1}. Zhong et al. [15, 82, 88, 89, 132] have investigated several half-discrete Hilbert-type inequalities with particular kernels.

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: ^∞₀𝑓(𝑥)^∞_𝑛=1𝑘_𝜆(𝑥,𝑛)𝑎_𝑛𝑑𝑥<𝑘𝜆₁𝑓_𝑝,𝜙‖𝑎‖_𝑞,𝜓, (13.3) where 𝑘(𝜆₁)=∫^∞₀𝑘_𝜆(𝑡,1)𝑡^𝜆1−1𝑑𝑡∈ℝ₊. This is an extension of the above particular result with the best constant factor 𝑘(𝜆₁) (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: ^∞₀𝑓(𝑥)^∞_𝑛=1𝑎_𝑛𝜋𝑥+𝑛𝑑𝑥<2𝑓₂‖𝑎‖₂,_∞0𝑓(𝑥)^∞_𝑛=1ln(𝑥/𝑛)𝑎_𝑛𝜋𝑥−𝑛𝑑𝑥<2²𝑓₂‖𝑎‖₂,_∞0𝑓(𝑥)^∞_𝑛=1𝑎_𝑛{}𝑓max𝑥,𝑛𝑑𝑥<4₂‖𝑎‖₂,_∞1𝑓(𝑥)^∞_𝑛=1|ln(𝑥/𝑛)|𝑎_𝑛{}𝑓max𝑥,𝑛𝑑𝑥<16₂‖𝑎‖₂,_∞1𝑓(𝑥)^∞_𝑛=1|ln(𝑥/𝑛)|𝑎_𝑛𝑥+𝑛𝑑𝑥<𝑐₀𝑓₂‖𝑎‖₂, (13.4)(13.5)(13.6)(13.7)(13.8) where 𝑐₀=8∑^∞_𝑛=1(−1)^𝑛/(2𝑛−1)²=7.3277⁺.

14. Concluding Remarks

(1)Many different kinds 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.(2)In his three books, Yang [116, 118, 119] presented 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.

Acknowledgments

The authors express their grateful thanks to Dr. Tibor Pogány and referees for many suggestions with recent references for improvement of the exposition of this paper. The work of the second author is supported by Guangdong Natural Science Foundation Grant no. 7004344.