A harmonic mean inequality for the digamma function and related results

  • Horst Alzer

    Waldbröl, Germany
  • Graham Jameson

    Lancaster University, UK

Abstract

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

Cite this article

Horst Alzer, Graham Jameson, A harmonic mean inequality for the digamma function and related results. Rend. Sem. Mat. Univ. Padova 137 (2017), pp. 203–209

DOI 10.4171/RSMUP/137-10