Some fundamental inequalities for Lerch transcendent function with positive terms by utilising certain classical results due to Hölder, Čebyšev, Grüss and others, are established. particular cases of interest Polylogarithm function, Hurwitz zeta Legendre chi also given.

Hurwitz formula for the generalized zeta function zeta (s,a) has been established under condition Re(s) = sigma < 0. Using the same contour integral, the proof proposed ill this paper allows the extension of this formula into the critical strip, 0 = a < 1. A similar result is obtained for the related function (1/Gamma (s)) integral (infinity)(0) x(s-1)(e(ax)/(1 + e(-x)))dx, 0 < a < 1.

Recently, Choi et al. 2008 have studied the q-extensions of the Apostol-Bernoulli and the ApostolEuler polynomials of order n and multiple Hurwitz zeta function. In this paper, we define Apostol’s type q-Euler numbers En,q,ξ and q-Euler polynomials En,q,ξ x . We obtain the generating functions of En,q,ξ and En,q,ξ x , respectively. We also have the distribution relation for Apostol’s type q-Eul...