A Half-Discrete Hardy-Hilbert-Type Inequality with a Best Possible Constant Factor Related to the Hurwitz Zeta Function

TL;DR

This paper establishes a new half-discrete Hardy-Hilbert-type inequality involving multiple parameters, with a best possible constant linked to the Hurwitz and Riemann zeta functions, using advanced real analysis techniques.

Contribution

It introduces a novel inequality with an optimal constant related to special zeta functions, expanding the theoretical framework of Hardy-Hilbert-type inequalities.

Findings

Derived a half-discrete inequality with a sharp constant

Connected the inequality to Hurwitz and Riemann zeta functions

Explored equivalent forms and operator expressions

Abstract

Using methods of weight functions, techniques of real analysis as well as the Hermite-Hadamard inequality, a half-discrete Hardy-Hilbert-type inequality with multi-parameters and a best possible constant factor related to the Hurwitz zeta function and the Riemann zeta function is obtained. Equivalent forms, normed operator expressions, their reverses and some particular cases are also considered.