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... In 1925, Hardy [1] gave a generalization of the Hilbert's inequality by introducing one pair of inequality (see [2]). As is known to us, the Hardy-Hilbert inequality plays an important role in analysis number theory, real analysis and divergent series theory (see [3]). ...

... Motivated by the Hardy-Hilbert inequality, in 1929, Mulholland [9] proposed a similar version of inequality (1), which contains the same best possible constant factor sin( / ) p as in (1), i.e., ...

... Volume 6, Issue 9, 9939-9954. 1 1 2 () ...

In this paper, we present a new reverse Mulholland-type inequality with multi-parameters and deal with its equivalent forms. Based on the obtained inequalities, the equivalent statements of the best possible constant factor related to several parameters are discussed. As an application, some interesting inequalities for double series are derived from the special cases of our main results.

... The boundedness of this operator has been heavily studied in the literature; in particular people have been very interested in the norm estimate of this operator and its siblings (see, e.g., the following and the references therein [1][2][3][4][5][6]). In [7], we considered a more general family of this operator for which we provided boundedness criteria and some sharp norm estimates. ...

... (i) The operator ⃗ , ⃗ is bounded from 1 1 ((0, ∞)) × ⋅ ⋅ ⋅ × ((0, ∞)) to ((0, ∞)). ...

... (i) The operator ⃗ , is bounded from 1 1 ((0, ∞)) × ⋅ ⋅ ⋅ × ((0, ∞)) to ((0, ∞)). ...

We consider two families of multilinear Hilbert-type operators for which we give exact relations between the parameters so that they are bounded. We also find the exact norm of these operators.

... If p > 1, 1 p + 1 q = 1, a m , b n ≥ 0, 0 < ∞ m=1 a p m < ∞, 0 < ∞ n=1 b q n < ∞, then we have the following Hardy-Hilbert inequality: ...

... with the best possible constant factor π sin(π /p) [1]. A more accurate form of (1) with the same best possible constant factor was given in [2, Theorem 323]: ...

... In this paper, using weight coefficients, a complex integral formula, and Hermite-Hadamard's inequality, we give the following extension of the reverse of (1) in the whole plane: If 0 < p < 1 (q < 0), 1 p + 1 q = 1, 0 < λ 1 , λ 2 < 1, λ 1 + λ 2 = λ ≤ 1, ξ , η ∈ [0, 1 2 ], a m , b n ≥ 0, ...

Using weight coefficients, a complex integral formula, and Hermite–Hadamard’s inequality, we give an extended reverse Hardy–Hilbert’s inequality in the whole plane with multiparameters and a best possible constant factor. Equivalent forms and a few particular cases are considered.

... where the constant factor π is the best. In 1925, by introducing a pair of conjugate parameters (p, q) ( 1 p + 1 q = 1, p > 1), Hardy [2] generalized (1) as follows: Ifã = {a m } ∈ l p , b = {b n } ∈ l q , then ...

... Define the series operator T by T(ã) n = ∞ ∑ m=1 1 m + n a m ,ã = {a m } ∈ l p , n = 1, 2, · · · It follows that T is a bounded operator on l p , and the operator norm of T is ||T|| = π/ sin(π/p). Therefore, it has important applications to study (2). By introducing an independent parameter λ, (2) has been extended to more general forms [3][4][5]. ...

... Therefore, it has important applications to study (2). By introducing an independent parameter λ, (2) has been extended to more general forms [3][4][5]. In 2020, ref. [6] considered the symmetric homogeneous kernel (min{m, n}) λ of λ-order, and obtained the Hilbert-type series inequality of the following form: If λ 1 + λ 2 = λ, a m ≥ 0, b m ≥ 0, then , and the equivalent conditions and expression formula for the best constant factor are discussed. ...

By introducing several independent parameters, according to the structural symmetry of quasi-homogeneous kernels and the Hilbert-type inequality, and using the weight function method, the parameter conditions of the optimal Hilbert-type n-multiple series inequality with quasi-homogeneous kernels are discussed, and several equivalent conditions and the expression formula of the best constant factor are obtained. As applications, some special symmetric inequalities are given.

... In [1], Hardy established that ...

... where δ and ω are measurable nonnegative functions such that 0 < ∞ 0 δ α (χ)dχ < ∞, 0 < ∞ 0 ω β (z)dz < ∞ and π/ sin(π/α) in (1), and (2) is the best value. In [2], Hardy showed that, if α > 1, β > 1, 1/α + 1/β ≥ 1 and 0 < λ = 2 − (1/α + 1/β) ≤ 1, then ...

... Some authors established the reverse Hölder inequalities, the reverse Young inequalities, and the reverse Hilbert inequalities by using the Specht's ratio function, see [8][9][10][11][12]. In particular, Zhao and Cheung [11] established the reverse Hölder inequalities by using the Specht's ratio function and proved that if ψ(ζ) and (ζ) are nonnegative continuous functions and ψ 1/α (ζ) 1 ...

This paper is interested in establishing some new reverse Hilbert-type inequalities, by using chain rule on time scales, reverse Jensen’s, and reverse Hölder’s with Specht’s ratio and mean inequalities. To get the results, we used the Specht’s ratio function and its applications for reverse inequalities of Hilbert-type. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. The new inequalities in special cases yield some recent relevance, which also provide new estimates on inequalities of these type.

... In 1925, by introducing one pair of conjugate exponents (p, q), Hardy [1] established a well-known extension of Hilbert's integral inequality as follows. ...

... [3], Theorem 350). In 1998, by introducing an independent parameter λ > 0, Yang proved an extension of Hilbert's integral inequality with the kernel 1 (x+y) λ (cf. [5,6]). ...

... [5,6]). In 2004, by introducing another pair of conjugate exponents (r, s), Yang [7] was able to estabish an extension of (1) with the kernel 1 x λ +y λ (λ > 0). In the paper [8], a further extension of (1) was proved along with the result of the paper [5] with the kernel 1 (x+y) λ . ...

In this paper, using weight functions as well as employing various techniques from real analysis, we establish a few equivalent conditions of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel. To prove our results, we also deduce a few equivalent conditions of two kinds of Hardy-type integral inequalities with a homogeneous kernel in the form of applications. We additionally consider operator expressions. Analytic inequalities of this nature and especially the techniques involved have far reaching applications in various areas in which symmetry plays a prominent role, including aspects of physics and engineering.

... where the constant factor / sin( / ) is the best possible one (cf. [1]). The more accurate form of (1) was given as follows (cf. ...

... where the constant factor /sin( / ) is still the best possible one. For = 0, inequality (2) reduces to (1). ...

By the use of weight coefficients and Hermite-Hadamard’s inequality, a new extension of Hardy-Hilbert’s inequality in the whole plane with multiparameters and a best possible constant factor is given. The equivalent forms, the operator expressions, and a few particular inequalities are considered.

... If p > 1, 1 p + 1 q = 1, a m , b n ≥ 0, 0 < ∞ m=1 a p m < ∞ and 0 < ∞ n=1 b q n < ∞, then we have the following Hardy-Hilbert's inequality (cf. [1]): ...

... If p > 1, 1 p + 1 q = 1, a m , b n ≥ 0, 0 < ∞ m=1 a p m < ∞ and 0 < ∞ n=1 b q n < ∞, then we have the following Hardy-Hilbert's inequality (cf. [1]): ...

By introducing independent parameters, applying the weight coefficients, and Hermite-Hadamard’s inequality, we give a more accurate Mulholland-type inequality in the whole plane with a best possible constant factor. Furthermore, the equivalent forms, the reverses, a few particular cases, and the operator expressions are considered.

... In [1], Hardy established that ...

... where ϕ, ψ ≥ 0 are measurable functions such that 0 < ∞ 0 ϕ l (ϑ)dϑ < ∞ and 0 < ∞ 0 ψ q (y)dy < ∞. The constant π/ sin(π/l) in both (1) and (2) is sharp. In [2], Hardy showed that if d > 1, q > 1, 1/d + 1/q ≥ 1 and 0 < λ = 2 − (1/d + 1/q) ≤ 1, then ...

This manuscript develops the study of reverse Hilbert-type inequalities by applying reverse Hölder inequalities on T. We generalize the reverse inequality of Hilbert-type with power two by replacing the power with a new power β,β>1. The main results are proved by using Specht’s ratio, chain rule and Jensen’s inequality. Our results (when T=N) are essentially new. Symmetrical properties play an essential role in determining the correct methods to solve inequalities.

... which was first proved by Hardy and Riesz [25] (also see [26, Chapter IX]). ...

We characterize the weights for the Stieltjes transform and the Calderón operator to be bounded on the weighted variable Lebesgue spaces \(L_w^{p(\cdot )}(0,\infty )\), assuming that the exponent function \({p(\cdot )}\) is log-Hölder continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator defined with respect to the basis of intervals \(\{ (0,b) : b>0\}\) on \((0,\infty )\). Our results extend those in Duoandikoetxea et al. (Indiana Univ Math J 62(3):891–910, 2013) for the constant exponent \(L^p\) spaces with weights. We also give two applications: the first is a weighted version of Hilbert’s inequality on variable Lebesgue spaces, and the second generalizes the results in Soria and Weiss (Indiana Univ Math J 43(1):187–204, 1994) for integral operators to the variable exponent setting.

... In 1925, Hardy [3] proved the following extension of Hilbert's integral inequality (cf. [4]): For p > 1, 1 p + 1 q = 1, f (x), g(y) 0, 0 < ∞ 0 f p (x)dx < ∞ and 0 < ∞ 0 g q (y)dy < ∞, the following Hardy-Hilbert inequality holds true: ...

We study some equivalent conditions of a reverse Hilbert-type integral inequality with a particular non-homogeneous kernel and a best possible constant factor related to the extended Hurwitz-zeta function. Some equivalent conditions of a reverse Hilbert-type integral inequality with the particular homogeneous kernel are deduced. We also consider some particular cases.

... which was first proved by G. Hardy and M. Riesz [25] (also see [26, Chapter IX]). ...

We characterize the weights for the Stieltjes transform and the Calder\'on operator to be bounded on the weighted variable Lebesgue spaces $L_w^{p(\cdot)}(0,\infty)$, assuming that the exponent function $p(\cdot)$ is log-H\"older continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator defined with respect to the basis of intervals $\{ (0,b) : b>0\}$ on $(0,\infty)$. Our results extend those in \cite{DMRO1} for the constant exponent $L^p$ spaces with weights. We also give two applications: the first is a weighted version of Hilbert's inequality on variable Lebesgue spaces, and the second generalizes the results in \cite{SW} for integral operators to the variable exponent setting.

... This inequality is called a Hardy-Hilbert type inequality. In [5], Hardy established the discrete version of (1.3), and he obtained the following inequality, if p > 1 with 1/p + 1/q = 1, {a m } ∞ m=1 and {b n } ∞ n=1 are nonnegative sequences such that ∞ m=1 a p m < ∞, and (1.4) where the constant π/ sin(π/p) is the best possible. In [7], Hardy considered a kernel K(x, y) = 1/ max(x, y), and he obtained the following inequality x p−2 f p (x)dx, (1.6) where Γ is the gamma function. ...

In this paper, we prove some new dynamic inequalities involving Hilbert and Hardy–Hilbert operators with kernels and use them to establish general forms of multiple Hilbert and Hardy–Hilbert type inequalities on time scales.

... In 1925, Hardy [1] gave an inequality as follows: ...

A reproducing kernel Hilbert space is a Hilbert space of complex-valued functions on a (non-empty) set Ω, which has the property that point evaluation is continuous on for all . Then the Riesz representation theorem guarantees that for every there is a unique element such that for all . The function is called the reproducing kernel of and the function is the normalized reproducing kernel in . The Berezin symbol of an operator A on a reproducing kernel Hilbert space is defined by The Berezin number of an operator A on is defined by The so-called Crawford number is defined by We also introduce the number defined by It is clear that By using the Hardy–Hilbert type inequality in reproducing kernel Hilbert space, we prove Berezin number inequalities for the convex functions in Reproducing Kernel Hilbert Spaces. We also prove some new inequalities between these numerical characteristics. Some other related results are also obtained.

... There have been a number of improvements and extensions on inequalities (1) and (2) (cf. [3][4][5][6][7][8][9][10]), which are important in the mathematical analysis and its applications (cf. [1,2,8]). ...

By using the theory of the local fractional calculus and the methods of weight function, a Hilbert-type fractional integral inequality with the kernel of Mittag-Leffler function and its equivalent form are given. Their constant factors are proved being the best possible, and its applications are discussed briefly.

... In 1925, Hardy gave an extension of (1) as follows [2]: If p > 1, 1 p + 1 q = 1, f (x) ≥ 0, satisfying 0 < ∞ 0 f p (x) dx < ∞, and g(y) ≥ 0, satisfying 0 < ∞ 0 g q (y) dy < ∞, then we have ...

Abstract By introducing independent parameters and interval variables, applying the weight functions and the technique of real analysis, an extended Hilbert’s integral inequality in the whole plane with parameters and a best possible constant factor is provided. The equivalent forms, the reverses, and the related homogeneous forms with particular parameters are considered. Meanwhile, an extended Hilbert’s integral operator in the whole plane is defined, and the operator expressions for the equivalent inequalities are obtained.

... Hardy [28] generalized the Hilbert inequality to L p case. For 1 < p < ∞, 1 p + 1 p = 1, a n and b n nonnegative sequences such that ∞ n=1 a p n < ∞ and ∞ n=1 b p n < ∞, it follows that Hardy et al. [29] gave more precise forms of the above inequalities ...

This article continues to study the linearized Chandrasekhar equation. We use the Hilbert-type inequalities to accurately calculate the norm of the Fredholm integral operator and obtain the exact range for the parameters of the linearized Chandrasekhar equation to ensure that there is a unique solution to the equation in \(L^p\) space. A series of examples that can accurately calculate the norm of Fredholm integral operator shows that the Chandrasekhar kernel functions do not need to meet harsh conditions. As the symbolic part of the Chandrasekhar kernel function and the non-homogeneous terms satisfy the exponential decay condition, we yield a normed convergence rate of the approximation solution in \(L^p\) sense, which adds new results to the theory of radiation transfer in astrophysics.

... where the constant π/ sin(π/p) is the best possible. In [7] Hardy proved the discrete version of (2) which is given by ...

In this paper, we will prove some new dynamic inequalities of Hilbert's type on time scales. Our results as special cases extend some obtained dynamic inequalities on time scales.and also contain some integral and discrete in- equalities as special cases. We prove our main results by using some algebraic inequalities, Hölder's inequality, Jensen's inequality and a simple consequence of Keller's chain rule on time scales.

... If p > 1, 1 p + 1 q = 1, a m , b n > 0, 0 < ∞ m=1 a p m < ∞, 0 < ∞ n=1 b q n < ∞, then we have the following discrete Hardy-Hilbert's inequality (cf. [3]): where, the constant factor π sin(π/p) is the best possible. Assuming that f (x), g(y) ≥ 0, satisfying 0 < ∞ 0 f p (x)dx < ∞ and 0 < ∞ 0 g q (y)dy < ∞, we have the following Hardy-Hilbert's integral inequality with the same best possible constant factor (cf. [4]): ...

By the use of Hermite-Hadamard’s inequality and weight functions, a new half-discrete Hilbert-type inequality in the whole plane with multi-parameters is given. The constant factor related to the gamma function is proved to be the best possible. The equivalent forms, two kinds of particular inequalities, and the operator expressions are considered.

... We call M to be the Hardy mean if H M < +∞; the number H M is called the Hardy constant of M. The definition of Hardy means was first introduced by Páles and Persson in [20] but it was developed since 1920s, when Hardy constants for Power means were given in a series of papers [9,13,4,10,11]; more details about interesting history of this result can be found in catching surveys [17,6] and in a recent book [12]. Term Hardy constant was introduced recently in [15]. ...

Each family ℳ of means has a natural, partial order (point-wise order), that is M ≤ N iff M ( x ) ≤ N ( x ) for all admissible x .
In this setting we can introduce the notion of interval-type set (a subset ℐ ⊂ ℳ such that whenever M ≤ P ≤ N for some M,N ∈ ℐ and P ∈ ℳ then P ∈ ℐ). For example, in the case of power means there exists a natural isomorphism between interval-type sets and intervals contained in real numbers. Nevertheless there appear a number of interesting objects for a families which cannot be linearly ordered.
In the present paper we consider this property for Gini means and Hardy means. Moreover, some results concerning L∞ metric among (abstract) means will be obtained.

... In 1925, Hardy [6] proved the following result, which is now very well known as the classical Hardy-Hilbert integral inequality. This states that for positive real numbers p, q with p > 1, 1 p + 1 q = 1, and functions f (x), g(y) ≥ 0, with ...

Making use of complex analytic techniques as well as methods involving weight functions, we study a few equivalent conditions of a Hilbert-type integral inequality with nonhomogeneous kernel and parameters. In the form of applications we deduce a few equivalent conditions of a Hilbert-type integral inequality with homogeneous kernel, and we additionally consider operator expressions.

... In [1] (p. 253), Hardy established that ...

Our work in this paper is based on the reverse Hölder-type dynamic inequalities illustrated by El-Deeb in 2018 and the reverse Hilbert-type dynamic inequalities illustrated by Rezk in 2021 and 2022. With the help of Specht’s ratio, the concept of supermultiplicative functions, chain rule, and Jensen’s inequality on time scales, we can establish some comprehensive and generalize a number of classical reverse Hilbert-type inequalities to a general time scale space. In time scale calculus, results are unified and extended. At the same time, the theory of time scale calculus is applied to unify discrete and continuous analysis and to combine them in one comprehensive form. This hybrid theory is also widely applied on symmetrical properties which play an essential role in determining the correct methods to solve inequalities. As a special case of our results when the supermultiplicative function represents the identity map, we obtain some results that have been recently published.

... In recent years, there were a lot of various refinements, generalizations, extensions, and applications of Hilbert's inequality which have seemed in the literature. Hilbert's discrete inequality and its integral formula ( [1], eorem 316) have been generalized in many trends (for example, see [2][3][4][5][6]). Lately, Pachpatte [7] proved new inequalities similar to those of Hilbert's inequality, namely, he proved that if h, l ≥ 1, A m � m s�1 a s ≥ 0, and B n � n t�1 b t ≥ 0, then ...

This paper is concerned with deriving some new dynamic Hilbert-type inequalities on time scales. The basic idea in proving the results is using some algebraic inequalities, Hölder’s inequality and Jensen’s inequality, on time scales. As a special case of our results, we will obtain some integrals and their corresponding discrete inequalities of Hilbert’s type.
1. Introduction
It is evident that the Hilbert-type inequalities outplay a major role in mathematics, for pattern complex analysis, numerical analysis, and qualitative theory of differential equations and their implementations. In recent years, there were a lot of various refinements, generalizations, extensions, and applications of Hilbert’s inequality which have seemed in the literature. Hilbert’s discrete inequality and its integral formula ([1], Theorem 316) have been generalized in many trends (for example, see [2–6]). Lately, Pachpatte [7] proved new inequalities similar to those of Hilbert’s inequality, namely, he proved that if h, l ≥ 1, , and , thenwhere
An integral analogue of (1) is given in the following result. Let h, l ≥ 1, , and , for and . Then,where
In 2001, Kim [8] gave some generalizations of (1) and (3) by introducing a parameter α > 0 aswhere h, l ≥ 1, , , and
An integral analogue of (5) is given in the following result. Let p, q ≥ 1, α > 0, , and , for , . Then,where
In 2009, Yang [9] gave another generalization of (1) and (3) by introducing parameter α > 1 and γ > 1 as follows. Let h, l ≥ 1, , and . Then,where
An integral analogue of (9) is given as follows. If h, l ≥ 1, α > 1, γ > 1, , and , for and , thenwhere
In [10], the authors deduced several generalizations of inequalities (1) and (3) on time scales, namely, they proved that if h and l ≥ 1 are real numbers, , , and η > 1, > 1 with , thenwhere
In [11], the authors gave some extensions of inequalities (5) and (7) on time scales. Minutely, they proved that if γ > 0 and h and l ≥ 1 are real numbers, , , and η > 1, > 1 with , thenwhere
Following this trend and to develop the study of dynamic inequalities on time scales, we will prove some new inequalities of Hilbert’s type on time scales, namely, we prove time scale versions of inequalities (9) and (11) on time scale . These inequalities can be considered as extensions and generalizations of some Hilbert-type inequalities proved in [10]. We also derive some inequalities on time scale as special cases.
2. Definitions and Basic Results
In this division, we will present some fundamental concepts and effects on time scales which will be beneficial for deducing our main results. The following definitions and theorems are referred from [12, 13].
Time scale is defined as a nonempty arbitrary locked subplot of real numbers . We define the forward jump operator asand the backward jump operator as
From the above two definitions, it can be stated that a point with is called right-scattered if σ(τ) > τ, right-dense if σ(τ) = τ, left-scattered if ρ(τ) < τ, and left-dense if ρ(τ) = τ. If has left-scattered maximum sm, then ; otherwise, . Finally, the graininess function for any is defined by
For a function , the delta derivative of χ at is defined as for each ɛ > 0, there is a neighborhood U of τ such that
Moreover, χ is called delta differentiable on if it is delta differentiable at every .
A function is called right-dense continuous (rd−continuous) as long as it is continuous at all right-dense points in , and its left-sided limits exist (finite) at all left-dense points in . The classes of real rd−continuous functions on an interval I will be denoted by . For θ, , the Cauchy integral of χΔ is defined as
Note that(a)If , then(b)If , then
In what follows, we will present Hölder’s inequality, Jensen’s inequality, and the power rules for integration on time scales.
Theorem 1. (Hölder’s inequality (see [14, 15])). Let . For , we havewhere η > 1 and > 1 with .
Theorem 2. (Jensen’s inequality (see [14, 16])). Suppose that and are nonnegative withIf is convex, then
Lemma 3. (see [17]). Let u, , and be nonnegative. If α ≥ 1, thenNow, we will present the formula that will reduce double integrals to single integrals which is the desired in [18].
Lemma 4. Let and . Then,assuming the integrals considered exist.
Lemma 5. (see [19]). Let r > 0, μq > 0, and Then,
3. Main Results
In this division, we will prove our main results. Throughout this section, we will assume that all functions are nonnegative and the integrals considered are assumed to exist. Also, we will assume that h and l ≥ 1 be real numbers and η > 1 and > 1 with .
Theorem 6. Let s, θ, and and and . Suppose that and are defined asThen, for and , we havewhere
Proof. By using inequality (27) (see Lemma 3), we see thatThen, we haveApplying Hölder’s inequality (1) on with indices η and η/(η − 1), we find thatand on the integral with indices and /( − 1), we find thatFrom (36) and (37), we getUsing inequality (29) of power means, we observe thatNow, by setting , , ω1 = 1/η, ω2 = 1/, and r = ω1 + ω2 in (39), we getSubstituting (40) into (38) yieldsDividing both sides of (41) by the last factor , we obtainIntegrating the above relation and applying Hölder’s inequality (1), we haveApplying Lemma 4 on (43) and using the fact that σ(n) ≥ n, we conclude thatwhich proves (31). This completes the proof.
Remark 1. Letting 1/η + 1/ = 1 in (31), we get Theorem 3.1 due to Saker et al. ([11], Theorem 3.1).
By using relations (22) and putting and t0 = 0 in Theorem 6, we get the following conclusion.
Corollary 7. Assume that f(ξ) and (ξ) are two nonnegative functions, and defineThen, for s ∈ (0, x) and θ ∈ (0, y), we havewherewhich was proved by Yang ([9], Theorem 3.1).
By using relations (23) and putting and t0 = 0 in Theorem 6, we get the following conclusion.
Corollary 8. Assume that {ai} and {bj} are two nonnegative sequences of real numbers, and defineThen,wherewhich was proved by Yang ([9], Theorem 2.1).
Remark 2. In Theorem 6, setting h = l = 1, we havewhere
Remark 3. In Remark 2, if , , and t0 = 0, then we get Remarks 2 and 5, respectively, due to Yang [9].
In the following theorems, we give a further generalization of (51) obtained in Remark 2. Before we give our result, we assume that there exist two functions Φ and Ψ which are real-valued, nonnegative, convex, and submultiplicative functions defined on . A function χ is a submultiplicative if χ(st) ≤ χ(s) χ(t) for s, t ≥ 0.
Theorem 9. Let s, θ, and , , , and h(τ) and l(ξ) be two positive functions defined for and . Suppose that F(s) and G(θ) are as defined in Theorem 6, and letThen, for and , we havewhere
Proof. According to Theorem 2 and the definition of function Φ, it is clear thatBy applying Hölder’s inequality (1) on (56), we find thatAnalogously,Thus, from (57) and (58), it can be acquired thatApplying (39) on the term , we getFrom (60), we observe thatIntegrating the above relation and using Hölder’s inequality (1) again with indices η, η/(η − 1) and , /( − 1), we find thatApplying Lemma 4 on (62) and using σ(n) ≥ n, we getwhich is (54). This completes the proof.
Remark 4. Letting 1/η + 1/ = 1 in (54), then we get Theorem 3.2 due to Saker et al. [11].
By using relations (22) and putting and t0 = 0 in Theorem 9, we get the following conclusion.
Corollary 10. Assume that f(s) and (θ) are two nonnegative functions and h(s) and l(θ) are two positive functions, and letThen, for s ∈ (0, x) and θ ∈ (0, y), we havewhereIt is clear that we can have the same inequality in [9], Theorem 3.2.
By using relations (23) and putting and t0 = 0 in Theorem 9, we get the following conclusion.
Corollary 11. Assume that {ai} and {bj} are two nonnegative sequences of real numbers and {hi} and {lj} are positive sequences, and defineThen,where
Remark 5. From inequality (39), we obtainIf we apply (70) on (31) in Theorem 6 and (54) in Theorem 9, then we get the following, respectively, inequalities:whereAlso,where
Remark 6. In Remark 5, if , , and t0 = 0, then we get Remarks 3 and 5, respectively, due to Yang [9].
The following theorems deal with slight variants of inequality (54) given in Theorem 9.
Theorem 12. Let s, θ, and , , and . DefineThen, for and , we havewhere
Proof. By assumption and using Jensen’s inequality (26), we see thatBy applying inequality (1) on (78) with indices η, η/(η − 1), we haveThis implies thatAnalogously,From (80) and (81), we getApplying elementary inequality (39) on the term , where , , ω1 = 1/η, ω2 = 1/, and r = ω1 + ω2, we getFrom (83), we haveTaking delta integrating on both sides of (84), first over s from t0 to x and then over θ from t0 to y, we find thatBy applying inequality (1) on (85) with indices η, η/(η − 1) and , /( − 1), we getApplying Lemma 4 on (86) and using the fact σ(n) ≥ n, we find thatThe proof is complete.
Remark 7. Letting 1/η + 1/ = 1 in (76), then we get Theorem 3.3 due to Saker et al. [11].
By using relations (22) and putting and t0 = 0 in Theorem 12, we get the following conclusion.
Corollary 13. Assume that f(s) and are nonnegative functions, and defineThen, for s ∈ (0, x) and θ ∈ (0, y), we havewherewhich is the same inequality in [9], Theorem 3.3.
By using relations (23) and putting and t0 = 0 in Theorem 12, we get the following conclusion.
Corollary 14. Assume that {ai} and {bj} are two nonnegative sequences of real numbers, and defineThen,wherewhich is the same inequality in [9], Theorem 2.3.
Theorem 15. Let s, θ, and , , , and h(ξ) and l(ξ) be two positive functions defined for and and H and L be as defined in Theorem 9, and letThen, for and , we havewhere
Proof. Using the hypotheses of Theorem 15 and Jensen’s inequality, we find thatBy applying inequality (1) on (97) with indices η, η/(η − 1), we haveFrom (98), we getAnalogously,From (99) and (100), we find thatApplying elementary inequality (39), we getThis implies thatTaking delta integrating on both sides of (103), first over s from t0 to x and then over θ from t0 to y, we obtainBy applying inequality (1) on (104) with indices η, η/(η − 1) and , /( − 1), we getApplying Lemma 4 and using σ(n) ≥ n, we getThis completes the proof.
Remark 8. Letting 1/η + 1/ = 1 in (95), then we get Theorem 3.4 due to Saker et al. [11].
By using relations (22) and putting and t0 = 0 in Theorem 15, we get the following conclusion.
Corollary 16. Assume that f(s) and are two nonnegative functions and h(s) and l(θ) are two positive functions, and defineThen, for s ∈ (0, x) and θ ∈ (0, y), we havewhereIt is clear that it is the same inequality in [9], Theorem 3.4.
By using relations (23) and putting and t0 = 0 in Theorem 15, we have the following conclusion.
Corollary 17. Assume that {ai} and {bj} are nonnegative sequences and {hi} and {lj} are positive sequences, and defineThen,wherewhich is the same inequality in [9], Theorem 2.4.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.

... Since both E. Landau and G. H. Hardy have contributed to the development of the inequality (1.1), so this inequality (1.1) sometimes called as Hardy-Landau inequality [15]. The 1925 articles of G. H. Hardy ([9], [10]) contain many interesting results. One of them is the following extension of Hardy's inequality (1.1). ...

In this current work, we revisit the recent improvement of the discrete Hardy's inequality in one dimension and establish an extended improved discrete Hardy's inequality with its optimality. We also study one-dimensional discrete Copson's inequality (E.T. Copson, \emph{Notes on a series of positive terms}, J. London Math. Soc., 2 (1927), 9-12.), and achieve an improvement of the same in a particular case. Further, we study some fundamental structures such as completeness, K\"{o}the-Toeplitz duality, separability, etc. of the sequence spaces which originated from the improved discrete Hardy and Copson inequalities in one dimension.

... In [1], Hardy established that ...

In this article, we establish some new generalizations of reversed dynamic inequalities of Hilbert-type via supermultiplicative functions by applying reverse Hölder inequalities with Specht’s ratio on time scales. We will generalize the inequalities by using a supermultiplicative function which the identity map represents a special case of it. Also, we use some algebraic inequalities such as the Jensen inequality and chain rule to prove the essential results in this paper. Our results (when T ≪ ℕ ) are essentially new.

... where Ξ and Υ are measurable functions such that, 0 < ∞ 0 Ξ 2 (τ)dτ < ∞ and 0 < ∞ 0 Υ 2 (y)dy < ∞. In 1925, by introducing one pair of conjugate exponents (p, q) with 1/p + 1/q = 1, Hardy [2] gave an extension of (1) as follows. If p, q > 1, β m , d n ≥ 0 such that 0 < ∑ ∞ m=1 β p m < ∞ and 0 < ∑ ∞ n=1 d q n < ∞, then ...

In this paper, we prove some new generalized inequalities of Hilbert-type on time scales nabla calculus by applying Hölder’s inequality, Young’s inequality, and Jensen’s inequality. Symmetrical properties play an essential role in determining the correct methods to solve inequalities.

... It is evident that Hilbert-type inequalities play a major role in mathematics, for complex pattern analysis, numerical analysis, qualitative theory of differential equations and their implementations. Hilbert's discrete inequality and its integral formula [1,Theorem 316] have been generalized in many ways (for example, see [2][3][4][5][6]). Lately, Pachpatte in [6], obtained the following inequality: if A q = q s=1 a s ≥ 0 and B n = n t=1 b t ≥ 0, for q = 1, 2, . . . ...

In this paper, we discuss some new Hilbert-type dynamic inequalities on time scales in two separate variables. We also deduce special cases, like some integral and their respective discrete inequalities.

... In [1], Hardy proved that ...

This study develops the study of reverse Hilbert-type inequalities on time scales where we can establish some new generalizations of reverse Hilbert-type inequalities via supermultiplicative functions on time scales by applying reverse Hölder inequalities. The main results will be proved by using Specht’s ratio, chain rule, and Jensen’s inequality. Our results (when T = ℕ ) are essentially new.

... For p > 1, 1 p + 1 q = 1, a m , b n > 0, the following discrete Hardy-Hilbert inequality (cf. [10], Theorem 315, and [4,11,36,40]) holds true: ...

A more accurate half-discrete Hilbert-type inequality in the whole plane with multi-parameters is established by the use of Hermite–Hadamard’s inequality and weight functions. Furthermore, some equivalent forms and some special types of inequalities and operator representations as well as reverses are considered.

... In one of the pioneering papers of Hardy [10] and [11], he proved the following (direct) inequality: ...

jats:p>In this note, we obtain a reverse version of the integral Hardy inequality on metric measure spaces. Moreover, we give necessary and sufficient conditions for the weighted reverse Hardy inequality to be true. The main tool in our proof is a continuous version of the reverse Minkowski inequality. In addition, we present some consequences of the obtained reverse Hardy inequality on the homogeneous groups, hyperbolic spaces and Cartan-Hadamard manifolds.
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... His proof was published by Weyl (1908). The first proof for general p, q > 1 was given by Hardy and Riesz [5]. Perhaps the best reference for the proof is [4]. ...

The Hilbert matrix $\mathcal{H}_{n,m} = (n+m+ 1)^{-1}$ has been extensively studied in previous literature. In this paper we look at generalized Hilbert operators arising from measures on the interval $[0, 1]$, such that the Hilbert matrix is obtained by the Lebesgue measure. We provide a necessary and sufficient condition for these operators to be bounded in $\ell^p$ and calculate their norm.

By using the Real and Functional Analysis and estimating the weight functions, we build two kinds of compositional Yang-Hilbert-type integral inequalities with the best possible constant factors. The equivalent forms and the reverses are also considered. Four kinds of compositional Yang-Hilbert-type integral operators are defined and the related composition formulas are given.

This chapter considers Hardy-Knopp type inequalities on an arbitrary time scale \( \mathbb{T} \). One-dimensional, two-dimensional and multidimensional versions Hardy-Knopp type inequalities are considered. Moreover, Hardy-Knopp type inequalities for several functions and refinement inequalities of Hardy-Knopp type with general kernels and Hardy-Knopp type on measure spaces are also discussed in this chapter. This chapter (with six sections) is organized as follows. In Sect. 7.1, we present a time scale version of inequalities of one dimension and then extend these inequalities to inequalities with two variables. In Sect. 7.2, we present some results for inequalities with two functions and in Sect. 7.3 we state and prove a number of Hardy-Knopp type inequalities on time scales using a convexity technique. In Sect. 7.4, we present some refinements of Hardy type inequalities with kernels. In Sect. 7.5, we generalize some delta-integral inequalities of Hardy type on time scales to diamond-α integrals.

By the use of Hermite–Hadamard’s inequality and weight functions, a half-discrete Hilbert-type inequality in the whole plane with the kernel of hyperbolic cotangent function and multi-parameters is given. The constant factor related to the Riemann zeta function is proved to be the best possible. The equivalent forms, two kinds of particular inequalities, the operator expressions and some equivalent reverses are considered.

By means of real analysis and weight functions, we obtain a few equivalent conditions of two kinds of Hardy-type integral inequalities with the non-homogeneous kernel and parameters. The constant factors related to the gamma function are proved to be the best possible. We also consider the operator expressions and some cases of homogeneous kernel.

In this paper, we prove some new dynamic inequalities of Hilbert type on time scales. From these inequalities, as special cases, we will formulate some special integral and discrete inequalities. The main results are proved using some algebraic inequalities, Hölder's inequality, Jensen's inequality and a chain rule on time scales.

Applying techniques of real analysis and weight functions, we study some equivalent conditions of two kinds of the reverse Hardy-type integral inequalities with a particular nonhomogeneous kernel. The constant factors are related to the Riemann zeta function and are proved to be best possible. In the form of applications, we deduce a few equivalent conditions of two kinds of the reverse Hardy-type integral inequalities with a particular homogeneous kernel. We also consider some corollaries as particular cases.

Las desigualdades han demostrado ser unas de las herramientas básicas para enfrentar múltiples problemas teórico-prácticos de la ciencia y la tecnología. En este artículo, guiados por la Teoría Dialéctica del Conocimiento, se realiza un estudio epistemológico de las condiciones de evolución y desarrollo de las desigualdades matemáticas, teniendo en cuenta su origen, sistematización y formalización.

By the use of techniques of real analysis and weight functions, we study some equivalent conditions of a reverse Hilbert-type integral inequality with the non-homogeneous kernel of hyperbolic cotangent function, related to the Riemann zeta function. Some equivalent conditions of a reverse Hilbert-type integral inequality with the homogeneous kernel are deduced. We also consider some particular cases.

In this paper, introducing multi-parameters and using properties of series, we prove a half-discrete Hilbert-type inequality in the whole plane with kernel in terms of the hyperbolic tangent function. The constant factor related to the Riemann zeta function and the gamma function is proved to be the best possible. In the form of applications, we also present equivalent forms, a few particular inequalities, operator expressions and reverses.

In this paper, we establish equivalent parameter conditions for the validity of multiple integral half-discrete Hilbert-type inequalities with the nonhomogeneous kernel G ( n λ 1 ∥ x ∥ m , ρ λ 2 ) G\left({n}^{{\lambda }_{1}}\parallel x{\parallel }_{m,\rho }^{{\lambda }_{2}}\hspace{-0.16em}) ( λ 1 λ 2 > 0 {\lambda }_{1}{\lambda }_{2}\gt 0 ) and obtain best constant factors of the inequalities in specific cases. In addition, we also discuss their applications in operator theory.

By the use of the way of real analysis and weight functions, we study some equivalent statements of Hardy-type integral inequality with the general nonhomogeneous kernel, related to another inequalities, as well as the parameters and the integrals of the kernel. As applications, a few equivalent stayements of Hardy-type integral inequality with the general homogeneous kernel are deduced. We also consider the other kind of integral inequality, the operator expressions, some corollaries and a few particular examples.

In the present paper, using weight coefficients and applying Hermite Hadamard’s inequality, we derive a new, more accurate reverse Hilbert-type inequality in the whole plane with multi-parameters involving the cosine and natural logarithm functions. The corresponding constant factor is proved to be the best possible. We additionally consider some equivalent forms and a few particular inequalities. As an application, the obtained results are compared with some previously known results and we show that these new results are more accurate than the earlier ones.

In the present paper, we prove some equivalent statements of a Hilbert-type integral inequality in the whole plane with intermediate variables. In our theorems, the constant factor is associated to the Hurwitz zeta function and we prove that it is the best possible. We also derive various special cases and applications.

By the use of the techniques of real analysis and the weight functions, a few equivalent statements of a general Hilbert-type integral inequality with the nonhomogeneous kernel related to another inequality, the parameters and the integral of kernel are obtained. The best possible constant factor is given. As a corollary, a few equivalent statements of a general Hilbert-type integral inequality with the homogeneous kernel and a best possible constant factor are deduced. Moreover, we also study the case of the reverses. The operator expressions, a few particular cases and some examples are considered.

Using weight functions, we obtain a half-discrete Hilbert-type inequality in the whole plane with the kernel of hyperbolic secant function and multi-parameters. The constant factor related to the Hurwitz zeta function is proved to be the best possible. We also consider equivalent forms, two kinds of particular inequalities, the operator expressions and some reverses.

In this paper, by using some classical operator means and classical operator inequalities, we investigate Berezin number of operators. In particular, we compare the Berezin number of some operator means of two positive operators. We also use some Hardy type inequalities to obtain a power inequality for the Berezin number of an operator. Moreover, by applying some inequalities for nonnegative Hermitian forms, some vector inequalities for n-tuple operators via Berezin symbols are established.

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