International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 834215, 14 pages
doi:10.1155/2009/834215
Some properties and regions of variability of affine harmonic mappings and affine biharmonic mappings
Sh. Chen1
, S. Ponnusamy2
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
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.