Certain Inequalities Involving the $q$-Deformed Gamma Function

This paper is inspired by the work of J. S'{a}ndor in 2006. In the paper, the authors establish some double inequalities involving the ratio $\frac{{\mathrm{\Gamma }}_q(x+1)}{{\mathrm{\Gamma }}_q(x+\frac{1}{2})}$, where ${\mathrm{\Gamma }}_q(x)$ is the $q$-deformation of the classical Gamma function denoted by $\mathrm{\Gamma }(x)$. The method employed in presenting the results makes use of Jackson’s $q$-integral representation of the $q$-deformed Gamma function. In addition, H"{o}lder’s inequality for the $q$-integral, as well as some basic analytical techniques involving the $q$-analogue of the psi function are used. As a consequence, $q$-analogues of the classical Wendel’s asymptotic relation are obtained. At the end, sharpness of the inequalities established in this paper is investigated.