Certain Conditions for Starlikeness of Analytic Functions of Koebe Type. (28th August 2011)

• Record Type: Journal Article
• Authors: Siregar, Saibah; Darus, Maslina
• Other Names: Zayed A. Academic Editor.
• Is Part Of: International journal of mathematics and mathematical sciences. Volume 2011(2011)
• Publication Date: 2011-08-28
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1155/2011/679704
• Languages: English
• ISSNs: 0161-1712

Abstract

Abstract : For α ≥ 0, λ > 0, we consider the M ( α, λ ) b of normalized analytic α - λ convex functions defined in the open unit disc U . In this paper, we investigate the class M ( α, λ ) b, that is, Re { (z f b ′ ( z ) / f b ( z )) [ 1 - α + α ( 1 - λ ) (z f b ′ ( z ) / f b ( z )) + α λ ( 1 + (z f b ′′ ( z ) / f b ′ ( z ) ) )] } > 0, with f b is Koebe type, that is, f b ( z ) : = z / ( 1 - 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, λ ) 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.