Inequalities for $\zeta(s)-\psi(1-s)$ related to a conjecture of Henry

TL;DR

This paper explores inequalities involving the difference between the Riemann zeta function and the digamma function, establishing properties like convexity and monotonicity, and deriving explicit boundary limits to support a conjecture of Henry.

Contribution

It introduces new inequalities and analytic properties of $ta(s)-ppa(1-s)$, advancing understanding related to Henry's conjecture.

Findings

Proves strict convexity and monotonicity of the difference function

Derives explicit boundary limits involving nd constants

Provides new inequalities supporting Henry's conjecture

Abstract

In this paper we investigate analytic inequalities related to a conjecture of Henry involving the difference between the Riemann zeta function and the digamma function. By treating $\zeta(s)-\psi(1-s)$ as a unified analytic object, we establish its strict convexity and monotonicity on suitable intervals. Moreover, we obtain explicit boundary limits of the derivative, expressed in terms of $\pi$, $\log (2\pi)$ and Stieltjes constants. These results lead to new inequalities for $\zeta(s)-\psi(1-s)$ and shed further light on the conjecture.