International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 834215, 14 pages
doi:10.1155/2009/834215
Research Article

Some Properties and Regions of Variability of Affine Harmonic Mappings and Affine Biharmonic Mappings

Sh. Chen,1 S. Ponnusamy,2 and X. Wang1

1Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
2Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India

Received 28 September 2009; Accepted 21 December 2009

Academic Editor: Narendra Kumar Govil

Copyright © 2009 Sh. Chen 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

We first obtain the relations of local univalency, convexity, and linear connectedness between analytic functions and their corresponding affine harmonic mappings. In addition, the paper deals with the regions of variability of values of affine harmonic and biharmonic mappings. The regions (their boundaries) are determined explicitly and the proofs rely on Schwarz lemma or subordination.