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Harmonic mappings of bounded boundary rotation


Authors: D. Bshouty, A. Lyzzaik and F. M. Sakar
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 30C45
DOI: https://doi.org/10.1090/proc/13796
Published electronically: September 28, 2017
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Abstract: The purpose of this paper is to investigate the valency of planar harmonic mappings of bounded boundary rotation of the open unit disc $ \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 $ \mathbb{D}$, proving an ameliorated statement of Theorem 2.1 and settling a modified version of Conjecture 4.1.


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Additional Information

D. Bshouty
Affiliation: Department of Mathematics, Technion, Haifa, Israel
Email: daoud@technion.ac.il

A. Lyzzaik
Affiliation: Astra Executive Consultant, Fahad Bin Sultan University, Tabuk, Saudi Arabia
Email: alyzzaik@gmail.com

F. M. Sakar
Affiliation: Department of Business Administration, Batman University, Batman, Turkey
Email: mugesakar@hotmail.com

DOI: https://doi.org/10.1090/proc/13796
Keywords: Planar harmonic mapping, harmonic mapping of bounded boundary rotation, close-to-convex function, multivalent function
Received by editor(s): August 3, 2016
Received by editor(s) in revised form: August 3, 2016, April 4, 2017, and April 13, 2017
Published electronically: September 28, 2017
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2017 American Mathematical Society