## 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)].

### MSC:

 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, $${}_pF_q$$

### Citations:

Zbl 1242.11065; Zbl 1228.11135
Full Text:

### References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.