On Some Properties of the Trigamma Function

Abstract

In 1974, Gautschi proved an intriguing inequality involving the gamma function $Γ$. Precisely, he proved that, for $z>0$, the harmonic mean of $Γ(z)$ and $Γ(1/z)$ can never be less than 1. In 2017, Alzer and Jameson extended this result to the digamma function $ψ$ by proving that, for $z>0$, the harmonic mean of $ψ(z)$ and $ψ(1/z)$ can never be less than $-γ$ where $γ$ is the Euler-Mascheroni constant. In this paper, our goal is to extend the results to the trigamma function $ψ'$. We prove among other things that, for $z>0$, the harmonic mean of $ψ'(z)$ and $ψ'(1/z)$ can never be greater than $π^2/6$.