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 $x\ne 1$, $$-\gamma <-\gamma H(x,1/x)<\frac{{\gamma }^2}{\psi (H(x,1/x))}<\psi (1/H(x,1/x))<H(\psi (x),\psi (1/x)),$$ $$H(\zeta (x),\zeta (1/x))<-2,$$ and for all $x\in (0,1)$ $$\zeta (1/2)<H(\zeta (x),\zeta (1-x))<-1,$$ $$\frac{\log 4}{1+\log 4}<H(\eta (x),\eta (1-x))<(1-\sqrt{2})\zeta (1/2).$$ Here, $\psi ={\Gamma }^{\prime }/\Gamma$ denotes the digamma function, $\gamma$ is Euler's constant, $\zeta$ is the Riemann's zeta function and $\eta$ is the Dirichlet's eta function.

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