A harmonic mean inequality for the $q-$gamma and $q-$digamma functions

Abstract

We prove among others results that the harmonic mean of $\Gamma_q(x)$ and $\Gamma_q(1/x)$ is greater than or equal to $1$
for arbitrary $x > 0$, and $q\in J$ where $J$ is a subset of $[0,+\infty)$. Also, we prove that there is $p_0\in(1,9/2)$, such that for $q\in(0,p_0)$, $\psi_q(1)$ is the minimum of the harmonic mean of $\psi_q(x)$ and $\psi_q(1/x)$ for $x > 0$ and for $q\in(p_0,+\infty)$, $\psi_q(1)$ is the maximum. Our results generalize some known inequalities due to Alzer and Gautschi.