International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 38089, 8 pages
doi:10.1155/IJMMS/2006/38089
Starlikeness and convexity of a class of analytic functions
1Faculty of Mechanical Engineering, Ss. Cyril and Methodius University, Karpoš II b.b., Skopje 1000, Macedonia
2Department of Mathematics Education, Faculty of Education, Başkent University, Bağlica Campus, Bağlica, Etimesgut, Ankara 06530, Turkey
Received 4 July 2006; Revised 4 August 2006; Accepted 10 August 2006
Copyright © 2006 Nikola Tuneski and Hüseyın Irmak. 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
Let 𝒜 be the class of analytic functions in the unit disk that
are normalized with f(0)=f′(0)−1=0 and let −1≤B<A≤1. In this paper we study the class Gλ,α={f∈𝒜:|(1−α+αzf″(z)/f′(z))/zf′(z)/f(z)−(1−α)|<λ,z∈𝒰},0≤α≤1, and give sharp sufficient conditions that embed it into the classes
S∗[A,B]={f∈𝒜:zf′(z)/f(z)≺(1+Az)/(1+Bz)}
and K(δ)={f∈𝒜:1+zf″(z)/f′(z)≺(1−δ)(1+z)/(1−z)+δ}, where “≺” denotes the usual subordination. Also, sharp upper
bound of |a2| and of the Fekete-Szegö functional |a3−μa22| is given for the class Gλ,α.