arXiv subjects
Publications and source records attributed to Edward Prempeh.
In this paper, the authors establish some inequalities involving the $q$-extension of the classical Gamma function. These inequalities provide bounds for certain ratios of the $q$-extended Gamma function. The procedure makes use of geometric convexity and monotonicy properties of certain functions associated with the $q$-extended Gamma function.
This paper is inspired by the work of J. Sándor in 2006. In the paper, the authors establish some double inequalities involving the ratio $ \frac{Γ_{q}(x+1)}{ Γ_{q} \left( x+\frac{1}{2}\right)}$, where $Γ_{q}(x)$ is the $q$-deformation of the classical Gamma function denoted by $Γ(x)$. The method employed in presenting the results makes use of Jackson's $q$-integral representation of the $q$-deformed Gamma function. In addition, Hölder'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.
In this paper, we present and prove some generalizations of some inequalities for the $p$-Gamma, $q$-Gamma and $k$-Gamma functions. Our approach makes use of the series representations of the psi, $p$-psi, $q$-psi and $k$-psi functions.