Study of some new one-parameter modifications of the Hardy-Hilbert integral inequality

By applying the way of real and functional analysis and estimating the weight functions, we build some lemmas and deduce some Hilbert-type and Hilbert-Hardy-type integral inequalities with the best possible constant factors. The equivalent forms, the reverses and the operator expressions are all considered. The composition formula of two Hilbert-Hardy-type integral operators and some examples are given.

This article deals with two fundamental topics in mathematical analysis: the formulation of integral expressions and the derivation of integral inequalities. In particular, it introduces new one-parameter integral formulas and inequalities of the logarithmic type, where the integrands involve the logarithmic function in one way or another. Among the results are weighted Hölder-type integral inequalities and two different forms of Hardy-Hilbert-type integral inequalities. These results are illustrated by various examples and accompanied by rigorous proofs.

In this paper, by introducing a new function with two parameters, we give another generalizations of the Hilbert's integral inequality with a mixed kernel k(x,y) = 1 A(x+y)+B|x y| and a best constant factors. As applications, some particular results with the best constant factors are considered.

In this paper, by introducing some parameters and using the way of weight function, a new integral inequality with a best constant factor is given, which is a relation between Hilbert's integral inequality and a Hilbert-type inequality. As applications, the equivalent form, the reverse forms and some particular inequalities are considered.

In this article, new Hardy-Hilbert-type integral inequalities are established. Our main result is based on a special inhomogeneous two-parameter kernel function. It is of the ratio power form, and has the property of involving a product term which perturbs the standard homogeneity property. We then use this result to derive new weighted integral norm inequalities and other Hardy-Hilbert-type integral inequalities. They are also defined with inhomogeneous kernel functions, but with innovative power and logarithmic forms. Some of them are obtained by treating an adjustable parameter as a variable and integrating with respect to it, which remains an original technique of proof. The article concludes with an at-tempt to unify some new and old Hardy-Hilbert-type integral inequalities. Due to the mathematical complexity, the optimality of the final result remains an open question, giving some new perspectives to a classical topic.

We give a new Hilbert‐type integral inequality with the best constant factor by estimating the weight function. And the equivalent form is considered.

On Hardy–Hilbert's Integral Inequality

By using the way of weight function, a new integral inequality with certain parameters and a best constant factor is proved which provides a relation of Hilbert’s integral inequality and a basic Hilbert-type integral inequality. Both the equivalent form as well as the reverse form are considered.KeywordsBasic Hilbert-type integral inequalityParameterWeight function

By introducing some parameters and a norm |x|α , x ∈ R+ , we give higherdimensional Hilbert’s and Hardy-Hilbert’s integral inequalities in non-conjugate case. Further, we prove that their constant factors are the best possible, in the conjugate case, when the parameters satisfy appropriate conditions. We also compare our results with some known results. Mathematics subject classification (2000): 26D15.

This catalogue of the HELP and HELP-type integral and series inequalities records the contributions made to this area of analytic inequalties from the years 1971–1996. The original HELP integral inequality came from the results of Hardy and Littlewood in one of their seminal papers, in this case written in 1932. The main analytic tools for the study of these inequalities are the properties of linear, ordinary, self-adjoint differential operators, and the properties of the Titchmarsh-Weyl / Hellinger-Nevanlinna m-coefficient and its ramifications. It is appropriate then, that this catalogue records some of the many distinguished contributions made to mathematical analysis in the first half of this century, by these named mathematicians. Likewise it is appropriate that this catalogue is dedicated to D.S. Mitrinovic whose contributions to the study and recording of analytic inequalities in the second half of this century, are now legendary.

Purpose. Improvement of the quasi-analytical method of nonlinear differential equation solution and its approbation with reference to beams of variable cross-section on the elastic base with two base factors.
 Research methods. Boundary conditions in the form of required number of correspondently transformed equations are added to the system of the linear algebraic equations which results from substitution of approximating function with constant factors (for example – power function) in the nonlinear differential equation and fixation of a set of variable values. The total number of the equations have to correspond to quantity of constant factors if the further solution will be carried out by an analytical method.
 Results. Deflection diagram of a trapezoid concrete beam with rectangular cross-section of variable height on the elastic base with two base factors has been calculated during approbation. Average solution error was equal to 0.06%. Distributions of the bending moments and normal stresses along the beam have been researched.
 Scientific novelty. The authors did not meet in literature such method of nonlinear differential equation solution.
 Practical value. The quasi-analytical method with realised consideration of boundary conditions that has been offered can be used for solution of differential equations of any order with various types of nonlinearity, including calculations of beams of variable cross-section on the elastic base.

A Hilbert‐type integral inequality with parametersαand (α,λ> 0) can be established by introducing a nonhomogeneous kernel function. And the constant factor is proved to be the best possible. And then some important and especial results are enumerated. As applications, some equivalent forms are studied.

This article introduces a new Hilbert-type integral inequality that is defined on the unit square and involves a singular integrand. A sharp upper bound is established using a change of variables based on the hyperbolic tangent function. As with classical Hilbert-type integral inequalities, the constant factor $\pi$ arises naturally. Furthermore, other inequalities are derived from the main result, including a new cosine-Hilbert-type integral inequality.

ABSTRACTBased on the local fractional calculus, by using the methods of weight function, a Hilbert-type fractal integral inequality and its equivalent form are given. Their constant factors are proved being the best possible, and its applications are discussed briefly.

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.