A harmonic mean inequality for the q-gamma function

In 1984, Kairies proved that for all positive real numbers x the geometric mean of $$\Gamma _q(x)$$ and $$\Gamma _q(1/x)$$ is greater than or equal to 1, that is, $$\begin{aligned} 1\le \sqrt{\Gamma _q(x)\Gamma _q(1/x)} \quad (0<q\ne 1), \end{aligned}$$ where $$\Gamma _q$$ denotes the q-gamma function. This result can be improved if $$q\in (0,1)$$ . We show that for all $$x>0$$ the harmonic mean of $$\Gamma _q(x)$$ and $$\Gamma _q(1/x)$$ is greater than or equal to 1, that is, $$\begin{aligned} 1\le \frac{2}{1/\Gamma _q(x)+1/\Gamma _q(1/x)} \quad (0<q<1) \end{aligned}$$ with equality if and only if $$x=1$$ .