Landau-type theorems for certain planar harmonic mappings or biharmonic mappings

Abstract

In this article, we first establish three new coefficient inequalities for planar bounded harmonic mappings in the unit disc U. Then we derive two versions of Landau-type theorems for bounded planar harmonic mappings or biharmonic mappings by applying these coefficient inequalities, which improve the related results of earlier authors. In particular, our results for certain bounded planar harmonic mappings such that | f(z)| ⩽ M for z ∈ U are sharp when M = 1.

Keywords:

• Landau-type theorem
• harmonic mapping
• biharmonic mapping

AMS Subject Classifications::

• Primary: 30C99
• Secondary: 30C62

Acknowledgements

This research is partly supported by the Natural Science Foundation of Fujian Province, China (Grant No. 2012J01005).