A Hilbert-type integral inequality under configuring free power and its applications

Qiong Liu

doi:10.1186/s13660-019-2039-1

Abstract

By using the method of weight function, the technique of real analysis, and the theory of special functions, a multi-parameter Hilbert-type integral inequality and its equivalent form are established, and their constant factors are proved to be the best possible. The expressions of operator with norm are given. As an application, relevant results in the references and some new inequalities are obtained by assigning some parameter values.

Highlights

If f, g : (0, ∞) → R are non-negative integrable functions, satisfying 0 < ∞ 0 f (x) dx < ∞, 0< ∞ 0 g2 (y) dy∞, the celebratedHilbert integral inequality is as follows:∞ ∞ f (x)g(y) dx dy < π ∞
In 2011, Yang gave an integral inequality of Hilbert type with exponential kernel as follows:
A series of Hilberttype integral inequalities with single kernels, mixed kernels, and compound kernels can be obtained by selecting appropriate parameter θ and other parameter values, so that the obtained results can be used more widely

Summary

Introduction

Hilbert integral inequality is as follows (see [1]):. 0 0 x+y where the constant factor π is the best possible. Hilbert integral inequality is as follows (see [1]):. In 2011, Yang gave an integral inequality of Hilbert type with exponential kernel as follows (see [16]): e–xyf (x)g dx dy. By using the method of weight function, the technique of real analysis, and the theory of special function, a Hilbert-type integral inequality and its equivalent form with the kernel as (min{1,xy})α (max{1,xy})β eγ xy are given, and their optimum constant factor in relation to Whittaker function and the application of the obtained results are briefly discussed. A series of Hilberttype integral inequalities with single kernels, mixed kernels, and compound kernels can be obtained by selecting appropriate parameter θ and other parameter values, so that the obtained results can be used more widely. The weight functions are defined by the following expressions:.

Main results

Operator expression with norm