International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Issue 18, Pages 38089, 8 p.
doi:10.1155/IJMMS/2006/38089
Starlikeness and convexity of a class of analytic functions
Nikola Tuneski1
and Hüseyin Irmak2
1Faculty of Mechanical Engineering, Ss. Cyril and Methodius University, Karpo {s} II b.b., Skopje 1000, Macedonia
2Department of Mathematics Education, Faculty of Education, Başkent University, Bağlica Campus, Bağlica, Etimesgut, Ankara 06530, Turkey
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.