Commun.Fac.Sci.Univ.Ank.Series A1 Volume 66, Number 2, Pages 130140 (2017) DOI: 10.1501/Commua1_0000000807 ISSN 13035991 Article electronically published on February 06, 2017 http://communications.science.ankara.edu.tr/index.php?series=A1 CONVEXITY PROPERTIES AND INEQUALITIES CONCERNING THE (p; k)-GAMMA FUNCTION KWARA NANTOMAH Abstract. In this paper, some convexity properties and some inequalities for the (p; k)-analogue of the Gamma function,  p;k (x) are given. In particular, a (p; k)-analogue of the celebrated Bohr-Mollerup theorem is given. Further- more, a (p; k)-analogue of the Riemann zeta function,  p;k (x) is introduced and some associated inequalities are derived. The established results provide the (p; k)-generalizations of some known results concerning the classical Gamma function. 1. Introduction In a recent paper [10], the authors introduced a (p; k)-analogue of the Gamma function dened for p 2 N, k> 0 and x 2 R + as  p;k (x)= Z p 0 t x1  1  t k pk  p dt (1.1) = (p + 1)!k p+1 (pk) x k 1 x(x + k)(x +2k) ::: (x + pk) (1.2) satisfying the basic properties  p;k (x + k)= pkx x + pk + k  p;k (x); (1.3)  p;k (ak)= p +1 p k a1  p (a); a 2 R +  p;k (k)=1: Received by the editors: August 16, 2016; Accepted: January 20, 2017. 2010 Mathematics Subject Classication. Primary: 33B15, Secondary: 33E50, 26A51. Key words and phrases. (p; k)-Gamma function, convex functions, Bohr-Mollerup theorem, (p; k)-Riemann zeta function, inequality. c 2017 Ankara University Communications de la FacultØ des Sciences de lUniversitØ dAnkara. SØries A1. Mathematics and Statistics. 130