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 $x\neq 1$, $$-\gamma<-\gamma H(x,1/x)<\frac{\gamma^2}{\psi\big(H(x,1/x)\big)}<\psi\Big(1/H(x,1/x)\Big)<H\Big(\psi(x), \psi(1/x)\Big),$$ $$H\Big(\zeta(x),\zeta(1/x)\Big)<-2,$$ 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.