Hilbert-type Inequalities Research Articles

A new Hilbert-type inequalities obtained via Cantor-type spherical coordinates

In this paper, we introduce a class of Hilbert-type inequalities on a fractal set [Formula: see text]. Establishing higher-dimensional Cantor-type spherical coordinates is an essential step in our results. We also impose the corresponding requirements that lead to the best constants obtained in the fractal Hilbert-type inequalities. Our results are applied by comparing them with previous results that are well known from the literature.

Construction of a class of half-discrete Hilbert-type inequalities in the whole plane

Abstract In this work, we first define two special sets of real numbers, and then, we construct a half-discrete kernel function where the variables are defined in the whole plane, and the parameters in the kernel function are limited to the newly constructed special sets. Estimate the kernel function in the whole plane by converting it to the first quadrant, and then, a class of new Hilbert-type inequality is established. Additionally, it is proved that the constant factor of the newly established inequality is the best possible. Furthermore, assigning special values to the parameters and using rational fraction expansion of cosecant function, some special results are presented at the end of this article.

On Some New Dynamic Hilbert-Type Inequalities Across Time Scales

In this article, we present some novel dynamic Hilbert-type inequalities within the framework of time scales T. We achieve this by utilizing Hölder’s inequality, the chain rule, and the mean inequality. As specific instances of our findings (when T=N and T=R), we obtain the discrete and continuous analogues of previously established inequalities. Additionally, we derive other inequalities for different time scales, such as T=qN0 for q>1, which, to the best of the authors’ knowledge, is a largely novel conclusion.

Discrete Local Fractional Hilbert-type Inequalities

The main objective of this paper is a study of some new discrete local fractional Hilbert-type inequalities. We apply our general results to homogeneous kernels. Also, the obtained results have the best possible constants.

A Weighted Generalization of Hardy–Hilbert-Type Inequality Involving Two Partial Sums

In this paper, we address Hardy–Hilbert-type inequality by virtue of constructing weight coefficients and introducing parameters. By using the Euler–Maclaurin summation formula, Abel’s partial summation formula, and differential mean value theorem, a new weighted Hardy–Hilbert-type inequality containing two partial sums can be proven, which is a further generalization of an existing result. Based on the obtained results, we provide the equivalent statements of the best possible constant factor related to several parameters. Also, we illustrate how the inequalities obtained in the main results can generate some new Hardy–Hilbert-type inequalities.

A reverse extended Hardy–Hilbert’s inequality with parameters

In this paper, by virtue of the symmetry principle, applying the techniques of real analysis and Euler–Maclaurin summation formula, we construct proper weight coefficients and use them to establish a reverse extended Hardy–Hilbert’s inequality with multi-parameters. Then, we obtain the equivalent forms and some equivalent statements of the best possible constant factor related to several parameters. Finally, we illustrate how the obtained results can generate some new reverse Hardy–Hilbert-type inequalities.

Refinements of Local Fractional Hilbert-Type Inequalities

Refinements of Local Fractional Hilbert-Type Inequalities

A half-discrete Hilbert-type inequality in the whole plane with the constant factor related to a cotangent function

In this work, by the introduction of some parameters, a new half-discrete kernel function in the whole plane is defined, which involves both the homogeneous and the nonhomogeneous cases. By employing some techniques of real analysis, especially the method of a weight function, a new half-discrete Hilbert-type inequality with the new kernel function, as well as its equivalent Hardy-type inequalities are established. Moreover, it is proved that the constant factors of the newly obtained inequalities are the best possible. Finally, assigning special values to the parameters, some new half-discrete Hilbert-type inequalities with special kernels are presented at the end of the paper.

Some Local Fractional Hilbert-Type Inequalities

The main purpose of this paper is to prove some new local fractional Hilbert-type inequalities. Our general results are applicable to homogeneous kernels. Furthermore, the best possible constants in terms of local fractional hypergeometric function are obtained. The obtained results prove that the employed method is very simple and effective for treating various kinds of local fractional Hilbert-type inequalities.

A More Accurate Half-Discrete Multidimensional Hilbert-Type Inequality Involving One Multiple Upper Limit Function

By means of the weight functions, the idea of introduced parameters, using the transfer formula and Hermite–Hadamard’s inequality, a more accurate half-discrete multidimensional Hilbert-type inequality with the homogeneous kernel as 1(x+||k−ξ||α)λ(x,λ>0) involving one multiple upper limit function is given, which is a new application of Hilbert-type inequalities. The equivalent conditions of the best possible constant factor related to several parameters are considered. The equivalent forms the operator expressions and some particular inequalities are obtained.

A class of half-discrete Hilbert-type inequalities in the whole plane involving some classical kernels

A class of half-discrete Hilbert-type inequalities in the whole plane involving some classical kernels

Refinements of local fractional Hilbert-type inequalities

UDC 517.5 We study the re nements of several well-known local fractional Hilbert-type inequalities by interpolating the Lebesgue norms of the local fractional Laplace transforms of the functions involved in the inequalities. As an application, the main results are compared with some our results previously known from the literature.

Some New Generalizations of Reverse Hilbert-Type Inequalities via Supermultiplicative Functions

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.

On a class of Hilbert-type inequalities in the whole plane involving some classical kernel functions

AbstractIn this paper, by the introduction of several parameters, we construct a new kernel function which is defined in the whole plane and includes some classical kernel functions. Estimating the weight functions with the techniques of real analysis, we establish a new Hilbert-type inequality in the whole plane, and the constant factor of the newly obtained inequality is proved to be the best possible. Additionally, by means of the partial fraction expansion of the tangent function, some special and interesting inequalities are presented at the end of the paper.

On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions

Abstract In this work, by the introduction of a new kernel function composed of exponent functions with several parameters, and using the method of weight coefficient, Hermite-Hadamard’s inequality, and some other techniques of real analysis, a more accurate half-discrete Hilbert-type inequality including both the homogeneous and non-homogeneous cases is established. Furthermore, by introducing the Bernoulli number and the rational fraction expansion of tangent function, some special and interesting Hilbert-type inequalities and their equivalent hardy-type inequalities are presented at the end of the paper.

On Some Important Dynamic Inequalities of Hardy–Hilbert-Type on Timescales

In this article, by using some algebraic inequalities, nabla Hölder inequalities, and nabla Jensen’s inequalities on timescales, we proved some new nabla Hilbert-type dynamic inequalities on timescales. These inequalities extend some known dynamic inequalities on timescales and unify some continuous inequalities and their corresponding discrete analogues. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

On Some Important Class of Dynamic Hilbert’s-Type Inequalities on Time Scales

In this important work, we discuss some novel Hilbert-type dynamic inequalities on time scales. The inequalities investigated here generalize several known dynamic inequalities on time scales and unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proved by using some algebraic inequalities, Hölder inequality, and Jensen’s inequality on time scales.

Some New Generalizations of Reverse Hilbert-Type Inequalities on Time Scales

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.

Important Study on the ∇ Dynamic Hardy–Hilbert-Type Inequalities on Time Scales with Applications

The main objective of the present article is to prove some new ∇ dynamic inequalities of Hardy–Hilbert-type on time scales. We present and prove very important generalized results with the help of the Fenchel–Legendre transform, submultiplicative functions, and Hölder’s and Jensen’s inequality on time scales. We obtain some well-known time scale inequalities due to Hardy–Hilbert inequalities. For some specific time scales, we further show some relevant inequalities as special cases: integral inequalities and discrete inequalities. Symmetry plays an essential role in determining the correct methods for solutions to dynamic inequalities

Some New Reverse Hilbert’s Inequalities on Time Scales

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.