Open Access

On univalent solutions of the biharmonic equation

Journal of Inequalities and Applications20052005:642561

DOI: 10.1155/JIA.2005.469

Received: 16 January 2004

Published: 31 October 2005

Abstract

We analyze the univalence of the solutions of the biharmonic equation. In particular, we show that if https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.469/MediaObjects/13660_2004_Article_1541_IEq1_HTML.gif is a biharmonic map in the form https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.469/MediaObjects/13660_2004_Article_1541_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.469/MediaObjects/13660_2004_Article_1541_IEq3_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.469/MediaObjects/13660_2004_Article_1541_IEq4_HTML.gif is harmonic, then https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.469/MediaObjects/13660_2004_Article_1541_IEq5_HTML.gif is starlike whenever https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.469/MediaObjects/13660_2004_Article_1541_IEq6_HTML.gif is starlike. In addition, when https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.469/MediaObjects/13660_2004_Article_1541_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.469/MediaObjects/13660_2004_Article_1541_IEq8_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.469/MediaObjects/13660_2004_Article_1541_IEq9_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.469/MediaObjects/13660_2004_Article_1541_IEq10_HTML.gif are harmonic, we show that https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.469/MediaObjects/13660_2004_Article_1541_IEq11_HTML.gif is locally univalent whenever https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.469/MediaObjects/13660_2004_Article_1541_IEq12_HTML.gif is starlike and https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.469/MediaObjects/13660_2004_Article_1541_IEq13_HTML.gif is orientation preserving.

Authors’ Affiliations

(1)
Department of Mathematics, American University of Sharjah

Copyright

© AbdulHadi et al. 2005