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International Journal of Mathematics and Mathematical Sciences
Volume 13, Issue 2, Pages 321-330

On close-to-convex functions of complex order

Department of Mathematics and Statistics, Bowling Green State University, Bowling Green 43403, OH, USA

Received 15 November 1988

Copyright © 1990 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The class S*(b) of starlike functions of complex order b was introduced and studied by M.K. Aouf and M.A. Nasr. The authors using the Ruscheweyh derivatives introduce the class K(b) of functions close-to-convex of complex order b, b0 and its generalization, the classes Kn(b) where n is a nonnegative integer. Here S*(b)K(b)=K0(b). Sharp coefficient bounds are determined for Kn(b) as well as several sufficient conditions for functions to belong to Kn(b). The authors also obtain some distortion and covering theorems for Kn(b) and determine the radius of the largest disk in which every fKn(b) belongs to Kn(1). All results are sharp.