A More Accurate Half-Discrete Hardy-Hilbert-Type Inequality with the Best Possible Constant Factor Related to the Extended Riemann-Zeta Function

TL;DR

This paper establishes a more precise half-discrete Hardy-Hilbert-type inequality involving the extended Riemann-zeta function, providing the best possible constant factor and various equivalent forms using advanced mathematical techniques.

Contribution

It introduces a new, more accurate inequality with the optimal constant factor related to the extended Riemann-zeta function, expanding the theoretical framework of Hardy-Hilbert inequalities.

Findings

Derived the inequality with the best possible constant factor

Presented equivalent forms and operator expressions

Explored reverses and particular cases

Abstract

By the method of weight coefficients, techniques of real analysis and Hermite-Hadamard's inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of the hyperbolic cosecant function with the best possible constant factor expressed in terms of the extended Riemann-zeta function is proved. The more accurate equivalent forms, the operator expressions with the norm, the reverses and some particular cases are also considered.