International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 3, Pages 469-474
doi:10.1155/S0161171295000597
Some properties of starlike functions with respect to symmetric-conjugate points
Hassoon Al-Amiri1
, Dan Coman2
and Petru T. Mocanu3
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
Abstract
Let A be tile class of all analytic functions in the unit disk U such that f(0)=f′(0)−1=0. A function f∈A is called starlike with respect to 2n symmetric-conjugate points if Rezf′(z)/fn(z)>0 for z∈U, where fn(z)=12n∑k=0n−1[ω−kf(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:A→A, I(f)=F where F(z)=c+1(g(z))c∫0zf(t)(g(t))c−1g′(t)dt, c>0 and g∈A is given are determined.