Harmonic mappings of bounded boundary rotation
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- by D. Bshouty, A. Lyzzaik and F. M. Sakar PDF
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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), 767โ774] and the recent work of T. Hayami [Complex Var. Elliptic Equ. 59 (2014), 1214โ1222].
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.
References
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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
- MR Author ID: 117325
- Email: alyzzaik@gmail.com
- F. M. Sakar
- Affiliation: Department of Business Administration, Batman University, Batman, Turkey
- Email: mugesakar@hotmail.com
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1113-1121
- MSC (2010): Primary 30C45
- DOI: https://doi.org/10.1090/proc/13796
- MathSciNet review: 3750223