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

We prove amongs others results that the harmonic mean of ${\mathrm{\Gamma }}_q(x)$ and ${\mathrm{\Gamma }}_q(1\mathrm{/}x)$ is greater than or equal to $1$ for arbitrary $x>0$ and $q\in J$where$J$isasubsetof$[0,+\infty)$\mathrm{.}Also,weprovethatforthereis$p_0\in(1,9/2)$,suchthatfor$q\in(0,p_0)$,$\psi_q(1)$ is the minimum of the harmonic mean of ${\psi }_q(x)$ and ${\psi }_q(1\mathrm{/}x)$ for $x>0$ and for $q\in (p_0,+\mathrm{\infty })$, ${\psi }_q(1)$ is the maximum. Our results generalize some known inequalities due to Alzer and Gautschi.