A mean value inequalities for the polygamma and zeta functions

TL;DR

This paper establishes new mean value inequalities involving the polygamma, zeta, and eta functions, extending known bounds and relations among these special functions for positive and fractional arguments.

Contribution

It proves novel inequalities linking harmonic means of special functions, providing deeper insights into their behavior and relationships beyond existing results.

Findings

Inequalities for the digamma function involving harmonic means.

Bounds for the Riemann zeta and eta functions using harmonic means.

Extensions of mean value inequalities to various special functions.

Abstract

A recently published result states inequalities of the harmonic mean of the digamma function. In this work, we prove among others results that for all positive real numbers $x\neq 1$, $$-\gamma<-\gamma H(x,1/x)<\frac{\gamma^2}{\psi\big(H(x,1/x)\big)}<\psi\Big(1/H(x,1/x)\Big)<H\Big(\psi(x), \psi(1/x)\Big),$$ $$H\Big(\zeta(x),\zeta(1/x)\Big)<-2,$$ and for all $x\in(0,1)$ $$\zeta(1/2)<H\Big(\zeta(x),\zeta(1-x)\Big)<-1,$$ $$\frac{\log 4}{1+\log 4}<H\Big(\eta(x),\eta(1-x)\Big)<(1-\sqrt 2)\zeta(1/2).$$ Here, $\psi=\Gamma'/\Gamma$ denotes the digamma function, $\gamma$ is Euler's constant, $\zeta$ is the Riemann's zeta function and $\eta$ is the Dirichlet's eta function.