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Abstract and Applied Analysis
Volume 2012, Article ID 721943, 32 pages
http://dx.doi.org/10.1155/2012/721943
Research Article

Univalent Logharmonic Mappings in the Plane

1Department of Mathematics, American University of Sharjah, P.O. Box 26666, Sharjah, UAE
2School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

Received 29 September 2011; Accepted 8 December 2011

Academic Editor: Saminathan Ponnusamy

Copyright © 2012 Zayid Abdulhadi and Rosihan M. Ali. 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

This paper surveys recent advances on univalent logharmonic mappings defined on a simply or multiply connected domain. Topics discussed include mapping theorems, logharmonic automorphisms, univalent logharmonic extensions onto the unit disc or the annulus, univalent logharmonic exterior mappings, and univalent logharmonic ring mappings. Logharmonic polynomials are also discussed, along with several important subclasses of logharmonic mappings.