A mean value inequalities for the polygamma and zeta 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 $x\neq 1$, $$-γ<-γH(x,1/x)<\frac{γ^2}{ψ\big(H(x,1/x)\big)}<ψ\Big(1/H(x,1/x)\Big)<H\Big(ψ(x), ψ(1/x)\Big),$$ $$H\Big(ζ(x),ζ(1/x)\Big)<-2,$$ and for all $x\in(0,1)$ $$ζ(1/2)<H\Big(ζ(x),ζ(1-x)\Big)<-1,$$ $$\frac{\log 4}{1+\log 4}<H\Big(η(x),η(1-x)\Big)<(1-\sqrt 2)ζ(1/2).$$ Here, $ψ=Γ'/Γ$ denotes the digamma function, $γ$ is Euler's constant, $ζ$ is the Riemann's zeta function and $η$ is the Dirichlet's eta function.