Table of Contents
ISRN Geometry
Volume 2011, Article ID 151474, 8 pages
http://dx.doi.org/10.5402/2011/151474
Research Article

Criteria for Strongly Starlike Functions

Department of Mathematics, Changshu Institute of Technology, Jiangsu, Changshu 215500, China

Received 5 April 2011; Accepted 11 May 2011

Academic Editors: A. Belhaj and A. A. Ungar

Copyright Β© 2011 Neng Xu. 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 𝑓(𝑧) be analytic in the unit disk π‘ˆ={π‘§βˆΆ|𝑧|<1} with 𝑓(0)=π‘“ξ…ž(0)βˆ’1=0 and (𝑓(𝑧)/𝑧)π‘“ξ…ž(𝑧)β‰ 0. By using the method of differential subordinations, we determine the largest number 𝛼(𝛽,πœ†,πœ‡,π‘š) such that, for some 𝛽,πœ†,πœ‡, and π‘š, the differential subordination πœ†(π‘§π‘“ξ…ž(𝑧)/𝑓(𝑧))1βˆ’πœ‡(1+(π‘§π‘“ξ…žξ…ž(𝑧)/π‘“ξ…ž(𝑧))βˆ’π‘§π‘“ξ…ž(𝑧)/𝑓(𝑧))+(π‘§π‘“ξ…ž(𝑧)/𝑓(𝑧))π‘šβ‰Ί(1+𝑧/1βˆ’π‘§)𝛼(𝛽,πœ†,πœ‡,π‘š)(π‘§βˆˆπ‘ˆ) implies π‘§π‘“ξ…ž(𝑧)/𝑓(𝑧)β‰Ί(1+𝑧/1βˆ’π‘§)𝛽. Some useful consequences of this result are also given.