International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 5, Pages 319-324
doi:10.1155/S0161171202007895

Sufficient conditions for starlikeness associated with parabolic region

V. Ravichandran,1 A. Gangadharan,1 and T. N. Shanmugam2

1Department of Mathematics and Computer Applications, Sri Venkateswara College of Engineering, Sriperumbudur 602 105, India
2Department of Mathematics, Anna University, Madras 600 025, India

Received 15 June 2001

Copyright © 2002 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.

Abstract

An analytic function f(z)=z+a n+1 z n+1+⋯, defined on the unit disk △={z:|z|<1}, is in the class S p if z f′(z)/f(z) is in the parabolic region Rew>|w−1|. This class is closely related to the class of uniformly convex functions. Sufficient conditions for function to be in S p are obtained. In particular, we find condition on λ such that the function f(z), satisfying (1−α)(f(z)/z) μ+αf′(z)(f(z)/z) μ−1≺1+λz, is in S p.