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