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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 $\Phi (z,s,a)$ 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 in [loc. cit], and by applying some Mathieu $(\alpha,\lambda)$-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 the last three authors [Appl. Math. Lett. 24, No. 8, 1473--1476 (2011; Zbl 1228.11135)].

11M35Hurwitz and Lerch zeta functions
30D15Special classes of entire functions; growth estimates
33B15Gamma, beta and polygamma functions
33C20Generalized hypergeometric series, ${}_pF_q$
Full Text: DOI
[1] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, North-holland mathematical studies 204 (2006) · Zbl 1092.45003
[2] Srivastava, H. M.; Saxena, R. K.; Pogány, T. K.; Saxena, R.: Integral and computational representations of the extended Hurwitz--lerch zeta function, Integral transforms spec. Funct. 22, 487-506 (2011) · Zbl 1242.11065
[3] Lin, S. -D.; Srivastava, H. M.: Some families of the Hurwitz--lerch zeta functions and associated fractional derivative and other integral representations, Appl. math. Comput. 154, 725-733 (2004) · Zbl 1078.11054 · doi:10.1016/S0096-3003(03)00746-X
[4] Goyal, S. P.; Laddha, R. K.: On the generalized zeta function and the generalized Lambert function, Ganita sandesh 11, 99-108 (1997) · Zbl 1186.11056
[5] Garg, M.; Jain, K.; Kalla, S. L.: A further study of general Hurwitz--lerch zeta function, Algebras groups geom. 25, 311-319 (2008) · Zbl 1210.11096
[6] Srivastava, H. M.; Choi, J.: Series associated with the zeta and related functions, (2001) · Zbl 1014.33001
[7] Choi, J.; Srivastava, H. M.: Certain families of series associated with the Hurwitz--lerch zeta function, Appl. math. Comput. 170, 399-409 (2005) · Zbl 1082.11052 · doi:10.1016/j.amc.2004.12.004
[8] Gupta, P. L.; Gupta, R. C.; Ong, S. -H.; Srivastava, H. M.: A class of Hurwitz--lerch zeta distributions and their applications in reliability, Appl. math. Comput. 196, 521-532 (2008) · Zbl 1131.62093 · doi:10.1016/j.amc.2007.06.012
[9] Lin, S. -D.; Srivastava, H. M.; Wang, P. -Y.: Some expansion formulas for a class of generalized Hurwitz--lerch zeta functions, Integral transforms spec. Funct. 17, 817-827 (2006) · Zbl 1172.11026 · doi:10.1080/10652460600926923
[10] Pogány, T. K.: Integral representation of Mathieu $(a,{\lambda})$-series, Integral transforms spec. Funct. 16, 685-689 (2005) · Zbl 1101.26018 · doi:10.1080/10652460500110297
[11] Qi, F.: Inequalities for Mathieu series, RGMIA res. Rep. coll. 4, No. 2, 187-193 (2001)
[12] Cerone, P.; Lenard, C. T.: On integral forms of generalized Mathieu series, J. inequal. Pure appl. Math. 4, No. 5 (2003) · Zbl 1072.26011
[13] Srivastava, H. M.; Tomovski, Ž.: Some problems and solutions involving Mathieu’s series and its generalizations, J. inequal. Pure appl. Math. 5, No. 2 (2004) · Zbl 1068.33032
[14] Pogány, T. K.; Srivastava, H. M.; Tomovski, Ž.: Some families of Mathieu a-series and alternating Mathieu a-series, Appl. math. Comput. 173, 69-108 (2006) · Zbl 1097.33016 · doi:10.1016/j.amc.2005.02.044
[15] Jankov, D.; Pogány, T. K.; Saxena, R. K.: Extended general Hurwitz--lerch zeta function as Mathieu $(a,{\lambda})$--series, Appl. math. Lett. 24, No. 8, 1473-1476 (2011) · Zbl 1228.11135 · doi:10.1016/j.aml.2011.03.040
[16] Pogány, T. K.; Srivastava, H. M.: Some Mathieu-type series associated with the fox--wright function, Comput. math. Appl. 57, 127-140 (2009) · Zbl 1165.33309 · doi:10.1016/j.camwa.2008.07.016
[17] , Applied mathematics series 55 (1964) · Zbl 0131.15908
[18] Elezović, N.; Giordano, C.; Pečarić, J.: The best bounds in gautschi’s inequality, Math. inequal. Appl. 3, 239-252 (2000) · Zbl 0947.33001