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International Journal of Mathematics and Mathematical Sciences
Volume 18, Issue 3, Pages 469-474

Some properties of starlike functions with respect to symmetric-conjugate points

1Department of Mathematics and Statistics, Bowling Gteen State Uiversity, Bowling Gteen 43403, Ohio, USA
2Departnent of Mathematics, University of Michigan, Ann Arbor 48109, MI, USA
3Faculty of Mathematics, Babes-Bolyai University, Cluj-Napoca 3400, Romania

Received 18 August 1993; Revised 3 February 1994

Copyright © 1995 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.


Let A be tile class of all analytic functions in the unit disk U such that f(0)=f(0)1=0. A function fA is called starlike with respect to 2n symmetric-conjugate points if Rezf(z)/fn(z)>0 for zU, where fn(z)=12nk=0n1[ωkf(ωkz)+ωkf(ωkz˜)¯], ω=exp(2πi/n]. This class is denoted by Sn*, and was studied in [1]. A sufficient condition for starlikeness with respect to symmetric-conjugate points is obtained. In addition, images of some subclasses of Sn* under the integral operator I:AA, I(f)=F where F(z)=c+1(g(z))c0zf(t)(g(t))c1g(t)dt,   c>0 and gA is given are determined.