On Some Properties of the Trigamma Function

In 1974, Gautschi proved an intriguing inequality involving the gamma function $\Gamma$. Precisely, he proved that, for $z>0$, the harmonic mean of $\Gamma(z)$ and $\Gamma(1/z)$ 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 $\psi(1/z)$ 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 $\psi'$. We prove among other things that, for $z>0$, the harmonic mean of $\psi'(z)$ and $\psi'(1/z)$ can never be greater than $\pi^2/6$.