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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 679704, 12 pages
http://dx.doi.org/10.1155/2011/679704
Research Article

Certain Conditions for Starlikeness of Analytic Functions of Koebe Type

1Faculty of Science and Biotechnology, Universiti Selangor, Bestari Jaya, 45600 Selangor Darul Ehsan, Malaysia
2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, 43600 Selangor Darul Ehsan, Malaysia

Received 1 May 2011; Revised 29 June 2011; Accepted 5 July 2011

Academic Editor: A. Zayed

Copyright © 2011 Saibah Siregar and Maslina Darus. 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 𝛼 β‰₯ 0 , πœ† > 0 , we consider the 𝑀 ( 𝛼 , πœ† ) 𝑏 of normalized analytic 𝛼 βˆ’ πœ† convex functions defined in the open unit disc π•Œ . In this paper, we investigate the class 𝑀 ( 𝛼 , πœ† ) 𝑏 , that is, R e { ( 𝑧 𝑓 ξ…ž 𝑏 ( 𝑧 ) / 𝑓 𝑏 ( 𝑧 ) ) [ 1 βˆ’ 𝛼 + 𝛼 ( 1 βˆ’ πœ† ) ( 𝑧 𝑓 ξ…ž 𝑏 ( 𝑧 ) / 𝑓 𝑏 ( 𝑧 ) ) + 𝛼 πœ† ( 1 + ( 𝑧 𝑓 𝑏 ξ…ž ξ…ž ( 𝑧 ) / 𝑓 ξ…ž 𝑏 ( 𝑧 ) ) ) ] } > 0 , with 𝑓 𝑏 is Koebe type, that is, 𝑓 𝑏 ( 𝑧 ) ∢ = 𝑧 / ( 1 βˆ’ 𝑧 𝑛 ) 𝑏 . The subordination result for the aforementioned class will be given. Further, by making use of Jack's Lemma as well as several differential and other inequalities, the authors derived sufficient conditions for starlikeness of the class 𝑀 ( 𝛼 , πœ† ) 𝑏 of 𝑛 -fold symmetric analytic functions of Koebe type. Relevant connections of the results presented here with those given in the earlier works are also indicated.