Rendiconti del Seminario Matematico della Università di Padova


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Volume 137, 2017, pp. 203–209
DOI: 10.4171/RSMUP/137-10

Published online: 2017-05-12

A harmonic mean inequality for the digamma function and related results

Horst Alzer[1] and Graham Jameson[2]

(1) Waldbröl, Germany
(2) Lancaster University, UK

We present some inequalities and a concavity property of the digamma function $\psi=\Gamma'/\Gamma$, 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 $x>0$.

Keywords: Digamma function, inequalities, concavity, harmonic mean, Euler’s constant

Alzer Horst, Jameson Graham: A harmonic mean inequality for the digamma function and related results. Rend. Sem. Mat. Univ. Padova 137 (2017), 203-209. doi: 10.4171/RSMUP/137-10