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Two-Sided Inequalities for the Extended Hurwitz-Lerch Zeta Function

Recently, Srivastava et al \cite{; ; SSPS}; ; unified and extended several interesting generali-zations of the familiar Hurwitz-Lerch Zeta function $\Phi(z, s, a)$ by introducing a Fox-Wright type generalized hypergeometric function in the kernel. For this newly introduced special function, two integral representations of different kind are investigated here by means a known result involving a Fox-Wright generalized hypergeometric function kernel, which was given earlier by Srivastava et al. \cite{; ; SSPS}; ; , and by applying some Mathieu $(a, \lambda)$-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. \cite{; ; JPS}; ; .

Extended Hurwitz-Lerch Zeta function; Fox-Wright ${;; };; _p\Psi_q^*$ function; Hypergeometric ${;; };; _pF_q$ function; Integral representations; Mathieu $(a; \lambda)$-series techniques; Psi (or Digamma) function; Euler-Mascheroni constant; Harmonic numbers; Two-sided bounding inequalities.

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