Abstract and Applied Analysis
Volume 2011 (2011), Article ID 901235, 10 pages
http://dx.doi.org/10.1155/2011/901235
Research Article

A Third-Order Differential Equation and Starlikeness of a Double Integral Operator

Rosihan M. Ali,1 See Keong Lee,1 K. G. Subramanian,1 and A. Swaminathan2

1School of Mathematical Sciences, Universiti Sains Malaysia (USM), Penang 11800, Malaysia
2Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247 667, India

Received 19 August 2010; Revised 2 January 2011; Accepted 10 January 2011

Academic Editor: Jean Pierre Gossez

Copyright © 2011 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

Functions ∑ 𝑓 ( 𝑧 ) = 𝑧 + ∞ 2 𝑎 𝑛 𝑧 𝑛 that are analytic in the unit disk and satisfy the differential equation 𝑓 ( 𝑧 ) + 𝛼 𝑧 𝑓 ' ' ( 𝑧 ) + 𝛾 𝑧 2 𝑓 ' ' ( 𝑧 ) = 𝑔 ( 𝑧 ) are considered, where 𝑔 is subordinated to a normalized convex univalent function ℎ . These functions 𝑓 are given by a double integral operator of the form ∫ 𝑓 ( 𝑧 ) = 1 0 ∫ 1 0 𝐺 ( 𝑧 𝑡 𝜇 𝑠 𝜈 ) 𝑡 − 𝜇 𝑠 − 𝜈 d s d 𝑡 with 𝐺 subordinated to ℎ . The best dominant to all solutions of the differential equation is obtained. Starlikeness properties and various sharp estimates of these solutions are investigated for particular cases of the convex function ℎ .