Two-sided inequalities for the extended Hurwitz–Lerch Zeta function

Recently, Srivastava et al. (2011) [2] unified and extended several interesting generalizations of the familiar Hurwitz–Lerch Zeta function Φ(z,s,a)Φ(z,s,a) by introducing a Fox–Wright type generalized hypergeometric function in the kernel. For this newly introduced special function, two integral representations of different kinds are investigated here by means of a known result involving a Fox–Wright generalized hypergeometric function kernel, which was given earlier by Srivastava et al. (2011) [2], and by applying some Mathieu (a,λ)-series techniques. Finally, by appealing to each of these two integral representations, two sets of two-sided bounding inequalities are proved, thereby extending and generalizing the recent work by Jankov et al. (2011) [15].