For q ∈ (0, 1) let the q-difference operator be defined as follows $$\partial _q f(z) = \frac{{f(qz) - f(z)}} {{z(q - 1)}} (z \in \mathbb{U}),$$ where \(\mathbb{U}\) denotes the open unit disk in a complex plane. Making use of the above operator the extended Ruscheweyh differential operator R _q ^λ f is defined. Applying R _q ^λ f a subfamily of analytic functions is defined. Several interesting properties of a defined family of functions are investigated.

In this paper, we investigate a new class of analytic functions defined by the convolution operator J _{s, b} (f) involving the Hurwitz–Lerch Zeta function, which was introduced recently by H. M. Srivastava and A. A. Attiya [Integral Transforms and Special Functions, 18, 2007, 207–216]. Coefficient inequalities, distortion theorems and other properties of functions belonging to this class are derived. We also indicate briefly the relevant connections of some of the results presented here with those that were proven in earlier investigations.

For ??(?,?], let Ra(?) denote the class of all normalized analytic functions in the open unit disk U satisfying the following differential subordination: f'(z)+1/2(1+ei?)z f''(z)<?(z) z ? U), where the function ?(z) is analytic in the open unit disk U such that ?(0)=1. In this paper, various integral and convolution characterizations, coefficient estimates and differential subordination results for functions belonging to the class R?(?) are investigated. The Fekete-Szeg? coefficient functional associated with the kth root transform [f(zk)]1/k of functions in R?(?) is obtained. A similar problem for a corresponding class R?,?(?) of bi-univalent functions is also considered. Connections with previous known results are pointed out.

This article deals with the ratio of normalized Mittag-Leffler function E _α,β (z) and its sequence of partial sums (E _α,β ) _m (z). Several examples which illustrate the validity of our results are also given.