International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 32, Pages 2053-2059
doi:10.1155/S0161171203207146
Subordination criteria for starlikeness and convexity
1Department of Mathematics, Faculty of Sciences, University of Urmia, Western Azerbaijan, Iran
2Department of Mathematical Sciences, Kent State University, Burton 44021-9500, OH, USA
Received 21 May 2002
Copyright © 2003 Rasoul Aghalary and Jay M. Jahangiri. 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
For functions p analytic in the open unit disc U={z:|z|<1} with the normalization p(0)=1, we consider the families 𝒫[A,−1], −1<A≤1, consisting of p such that p(z) is subordinate to (1+Az)/(1−z) in U and 𝒫(1,b), b>0,
consisting of p, which have the disc formulation |p−1|<b in U. We then introduce subordination criteria for the choice of
p(z)=zf′(z)/f(z), where f is analytic in U and normalized by f(0)=f′(0)−1=0. We also obtain starlikeness and convexity
conditions for such functions f and consequently extend and, in
some cases, improve the corresponding previously known results.