ABSTRACT. For q ∈ (0, 1) let the q-difference operator be defined as followswhere U denotes the open unit disk in a complex plane. Making use of the above operator the extended Ruscheweyh differential operator R λ q f is defined. Applying R λ q f a subfamily of analytic functions is defined. Several interesting properties of a defined family of functions are investigated.

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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.