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Certain subclass of analytic functions defined by means of differential subordination

Srivastava H.M. (University of Victoria, Department of Mathematics and Statistics, Victoria, British Columbia, Canada + China Medical University, Taichung, Taiwan, Republic of China)
Răducanu Dorina (Transilvania University of Braşov, Faculty of Mathematics and Computer Science, Braşov, Romania)
Zaprawa Paweł (Lublin University of Technology, Department of Mathematics, Lublin, Poland)

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

Keywords: Analytic functions, Univalent functions, Bi-Univalent functions, Differential subordination, Fekete-Szegö problem