International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 8, Pages 1171-1187
doi:10.1155/IJMMS.2005.1171

On the generalized Roper-Suffridge extension operator in Banach spaces

Ming-Sheng Liu1 and Yu-Can Zhu2

1School of Mathematical Science, South China Normal University, Guangzhou, Guangzhou 510631, China
2Department of Mathematics, Fuzhou University, Fuzhou, Fujian 350002, China

Received 12 July 2004; Revised 28 February 2005

Copyright © 2005 Ming-Sheng Liu and Yu-Can Zhu. 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

The generalized Roper-Suffridge extension operator in Banach spaces is introduced. We prove that this operator preserves the starlikeness on some domains in Banach spaces and does not preserve convexity in some cases. Furthermore, the growth theorem and covering theorem of the corresponding mappings are given. Some results of Roper and Suffridge and Graham et al. in ℂn are extended to Banach spaces.