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

Abstract

We prove amongs 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 for 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.