A Hilbert-Type Integral Inequality in the Whole Plane Related to the Hypergeometric Function and the Beta Function

TL;DR

This paper introduces a new Hilbert-type integral inequality over the entire plane involving hypergeometric and beta functions, establishing the optimal constant and exploring various related inequalities and applications.

Contribution

It presents a novel Hilbert-type inequality with a non-homogeneous kernel, linking it to special functions and demonstrating its optimality and applications.

Findings

Established the best possible constant involving hypergeometric and beta functions.

Derived equivalent forms and reverses of the inequality.

Applied the inequality to Hardy-type inequalities and operator expressions.

Abstract

A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel and parameters is given. The constant factor related to the hypergeometric function and the beta function is proved to be the best possible. As applications, equivalent forms, the reverses, some particular examples, two kinds of Hardy-type inequalities, and operator expressions are considered.