A mean value inequalities for the polygamma and zeta functions

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≠ 1, -γ<-γ H(x,1/x)<γ²/ψ(H(x,1/x))<ψ(1/H(x,1/x))<H(ψ(x),ψ(1/x)), H(ζ(x),ζ(1/x))<-2, and for all x∈(0,1) ζ(1/2)<H(ζ(x),ζ(1-x))<-1, log 4/1+log 4<H(η(x),η(1-x))<(1-√ 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.