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)].


11M35 Hurwitz and Lerch zeta functions
30D15 Special classes of entire functions of one complex variable and growth estimates
33B15 Gamma, beta and polygamma functions
33C20 Generalized hypergeometric series, \({}_pF_q\)
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[1] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., ()
[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
[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), Kluwer Academic Publishers Dordrecht, Boston, London · 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
[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
[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
[10] Pogány, T.K., Integral representation of Mathieu \((\mathbf{a}, \lambda)\)-series, Integral transforms spec. funct., 16, 685-689, (2005) · Zbl 1101.26018
[11] Qi, F., Inequalities for Mathieu series, RGMIA res. rep. coll., 4, 2, 187-193, (2001), Article 3
[12] Cerone, P.; Lenard, C.T., On integral forms of generalized Mathieu series, J. inequal. pure appl. math., 4, 5, (2003), Article 100, 1-11 (electronic) · 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, 2, (2004), Article 45, 1-13 (electronic) · Zbl 1068.33032
[14] Pogány, T.K.; Srivastava, H.M.; Tomovski, Ž., Some families of Mathieu a-series and alternating Mathieu {\bfa}-series, Appl. math. comput., 173, 69-108, (2006) · Zbl 1097.33016
[15] Jankov, D.; Pogány, T.K.; Saxena, R.K., Extended general hurwitz – lerch zeta function as Mathieu \((\boldsymbol{a}, \mathbf{\lambda})\)-series, Appl. math. lett., 24, 8, 1473-1476, (2011) · Zbl 1228.11135
[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
[17] (), Reprinted by Dover Publications, New York, 1965
[18] Elezović, N.; Giordano, C.; Pečarić, J., The best bounds in gautschi’s inequality, Math. inequal. appl., 3, 239-252, (2000) · Zbl 0947.33001
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