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Two-sided inequalities for the extended Hurwitz-Lerch zeta function. (English) Zbl 1228.11137
Summary: Recently, the authors [Integral Transforms Spec. Funct. 22, No. 7, 487–506 (2011; Zbl 1242.11065)] unified and extended several interesting generalizations of the familiar Hurwitz-Lerch zeta function ${\Phi }\left(z,s,a\right)$ 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 in [loc. cit], and by applying some Mathieu $\left(\alpha ,\lambda \right)$-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 the last three authors [Appl. Math. Lett. 24, No. 8, 1473–1476 (2011; Zbl 1228.11135)].
##### MSC:
 11M35 Hurwitz and Lerch zeta functions 30D15 Special classes of entire functions; growth estimates 33B15 Gamma, beta and polygamma functions 33C20 Generalized hypergeometric series, ${}_{p}{F}_{q}$
##### References:
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