A harmonic mean inequality for the digamma function and related results

Abstract

We present some inequalities and a concavity property of the digamma function $\psi ={\Gamma }^{\prime }/\Gamma$, where $\Gamma$ denotes Euler's gamma function. In particular, we offer a new characterization of Euler's constant $\gamma =0.57721\dots$. We prove that $-\gamma$ is the minimum of the harmonic mean of $\psi (x)$ and $\psi (1/x)$ for $x>0$.