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Note on a theorem of Hilbert concerning series of positive term

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