Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 175369, 11 pages
doi:10.1155/2010/175369
Research Article

Some Starlikeness Criterions for Analytic Functions

Gejun Bao,1 Lifeng Guo,1 and Yi Ling2

1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2Department of Mathematical Science, Delaware State University, Dover, DE 19901, USA

Received 26 October 2010; Accepted 16 December 2010

Academic Editor: Vijay Gupta

Copyright © 2010 Gejun Bao et al. 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

We determine the condition on 𝛼 , 𝜇 , 𝛽 , and 𝜆 for which | ( 1 − 𝛼 ) ( 𝑧 / 𝑓 ( 𝑧 ) ) 𝜇 + 𝛼 ( 𝑧 𝑓 ( 𝑧 ) / 𝑓 ( 𝑧 ) ) ( 𝑧 / 𝑓 ( 𝑧 ) ) 𝜇 − 1 | < 𝜆 implies 𝑓 ( 𝑧 ) ∈ 𝑆 ∗ ( 𝛽 ) , where 𝑆 ∗ ( 𝛽 ) is the class of starlike functions of order 𝛽 . Some results of Obradović and Owa are extended. We also obtain some new results on starlikeness criterions.