Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 712328, 18 pages
doi:10.1155/2008/712328
Research Article

Subordination and Superordination on Schwarzian Derivatives

Rosihan M. Ali,1 V. Ravichandran,2 and N. Seenivasagan3

1School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800 Penang, Malaysia
2Department of Mathematics, University of Delhi, Delhi 110 007, India
3Department of Mathematics, Rajah Serfoji Government College, Thanjavur 613 005, India

Received 4 September 2008; Accepted 30 October 2008

Academic Editor: Paolo Ricci

Copyright © 2008 Rosihan M. Ali 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

Let the functions q1 be analytic and let q2 be analytic univalent in the unit disk. Using the methods of differential subordination and superordination, sufficient conditions involving the Schwarzian derivative of a normalized analytic function f are obtained so that either q1(z)≺zf′(z)/f(z)≺q2(z) or q1(z)≺1+zf′′(z)/f′(z)≺q2(z). As applications, sufficient conditions are determined relating the Schwarzian derivative to the starlikeness or convexity of f.