Convexity properties and inequalities regarding the q-Gamma function

Abstract

This paper is motivated by the work of K. Nantomah in 2017. In the paper, some convexity properties and some inequalities for the (p, k)-analogue of the Gamma function,〖 Γ〗_(p,k) (x) were given. The method engaged in displaying the result makes use of the (p, k)-analogue of the Gamma function. In addition, H(ol) ̈der’s integral inequality, Young’s inequality and some basic definitions of a convex function were used. As a result, the (p, k)-generalization of some known outcomes concerning the classical gamma function was specified. The fundamental objective of this paper is to ascertain some convexity properties and some inequalities regarding the q-analogue of the Gamma function, Γ_q (x). First, utilizing similar techniques as K. Nantomah, the convexity property of the q-Gamma function was demonstrated. Next, exploiting Young’s inequality, some inequalities regarding the q-Gamma function were substantiated. At the end, the q-analogue of some accepted results concerning the classical Gamma function was proven.