A Multidimensional Hilbert-Type Integral Inequality Related to the Riemann Zeta Function

TL;DR

This paper establishes a new multidimensional Hilbert-type integral inequality involving the Riemann zeta function, providing equivalent forms, reverses, and operator-related results using real analysis techniques.

Contribution

It introduces a novel multidimensional inequality with a sharp constant linked to the Riemann zeta function, expanding the theoretical framework of integral inequalities.

Findings

Derived a multidimensional Hilbert-type inequality with a sharp constant

Presented equivalent representations and reverses of the inequality

Explored operator expressions related to the inequality

Abstract

In this paper,using methods of weight functions and techniques of real analysis, we provide a multidimensional Hilbert-type integral inequality with a homogeneous kernel of degree 0 as well as a best possible constant factor related to the Riemann zeta function. Some equivalent representations and certain reverses are obtained. Furthermore, we also consider operator expressions with the norm and some particular results.