Journal of Inequalities and Applications
Volume 2005 (2005), Issue 5, Pages 469-478
doi:10.1155/JIA.2005.469
On univalent solutions of the biharmonic equation
Department of Mathematics, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates
Received 16 January 2004
Copyright © 2005 Z. AbdulHadi 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 analyze the univalence of the solutions of the biharmonic
equation. In particular, we show that if F is a biharmonic map
in the form F(z)=r2G(z), |z|<1, where G is harmonic,
then F is starlike whenever G is starlike. In addition, when F(z)=r2G(z)+K(z), |z|<1, where G and K are harmonic,
we show that F is locally univalent whenever G is starlike and K is orientation preserving.