On Some Properties of the Trigamma Function

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