Abstract and Applied Analysis
Volume 2011 (2011), Article ID 865496, 14 pages
http://dx.doi.org/10.1155/2011/865496
Research Article

A New Roper-Suffridge Extension Operator on a Reinhardt Domain

Jianfei Wang and Cailing Gao

Department of Mathematics and Physics, Information Engineering, Zhejiang Normal University, Zhejiang, Jinhua 321004, China

Received 5 July 2011; Revised 28 September 2011; Accepted 5 October 2011

Academic Editor: Sung Guen Kim

Copyright © 2011 Jianfei Wang and Cailing Gao. 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

We introduce a new Roper-Suffridge extension operator on the following Reinhardt domain Ω 𝑛 , 𝑝 2 , … , 𝑝 𝑛 = { 𝑧 ∈ ℂ 𝑛 ∶ | 𝑧 1 | 2 + ∑ 𝑛 𝑗 = 2 | 𝑧 𝑗 | 𝑝 𝑗 < 1 } given by 𝐹 ( 𝑧 ) = ( 𝑓 ( 𝑧 1 ) + 𝑓 ′ ( 𝑧 1 ) ∑ 𝑛 𝑗 = 2 𝑎 𝑗 𝑧 𝑝 𝑗 𝑗 , ( 𝑓 ′ ( 𝑧 1 ) ) 1 / 𝑝 2 𝑧 2 , … , ( 𝑓 ′ ( 𝑧 1 ) ) 1 / 𝑝 𝑛 𝑧 𝑛 ) , where 𝑓 is a normalized locally biholomorphic function on the unit disc 𝐷 , 𝑝 𝑗 are positive integer, 𝑎 𝑗 are complex constants, and 𝑗 = 2 , … , 𝑛 . Some conditions for 𝑎 𝑗 are found under which the operator preserves almost starlike mappings of order 𝛼 and starlike mappings of order 𝛼 , respectively. In particular, our results reduce to many well-known results when all 𝛼 𝑗 = 0 .