Harmonic mappings of bounded boundary rotation

Abstract

The purpose of this paper is to investigate the valency of planar harmonic mappings of bounded boundary rotation of the open unit disc D . \mathbb {D}. The paper is motivated by the earlier work of the first two authors [Complex Analysis Oper. Theory 5 (2011), 767–774] and the recent work of T. Hayami [Complex Var. Elliptic Equ. 59 (2014), 1214–1222]. First, the authors give a counterexample showing that both the main result of Hayami, Theorem 2.1, and the related conjecture, Conjecture 4.1, are false. Second, the authors give a valency criterion for planar harmonic mappings of bounded boundary rotation of D \mathbb {D} , proving an ameliorated statement of Theorem 2.1 and settling a modified version of Conjecture 4.1.