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CLASSIFICATION OF UNIVALENT HARMONIC MAPPINGS ON THE UNIT DISK WITH HALF-INTEGER COEFFICIENTS

Part of: Geometric function theory

Published online by Cambridge University Press:  12 November 2014

SAMINATHAN PONNUSAMY  and
JINJING QIAO
SAMINATHAN PONNUSAMY*
Affiliation:
Indian Statistical Institute (ISI), Chennai Centre SETS (Society for Electronic Transactions and Security), MGR Knowledge City, CIT Campus, Taramani, Chennai 600 113, India email samy@isichennai.res.in, samy@iitm.ac.in
JINJING QIAO
Affiliation:
Department of Mathematics, Hebei University, Baoding, Hebei 071002, PR China email mathqiao@126.com
*
✉samy@isichennai.res.in, samy@iitm.ac.in
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Abstract

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Let ${\mathcal{S}}$ denote the set of all univalent analytic functions $f$ of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ on the unit disk $|z|<1$. In 1946, Friedman [‘Two theorems on Schlicht functions’, Duke Math. J.13 (1946), 171–177] found that the set ${\mathcal{S}}_{\mathbb{Z}}$ of those functions in ${\mathcal{S}}$ which have integer coefficients consists of only nine functions. In a recent paper, Hiranuma and Sugawa [‘Univalent functions with half-integer coefficients’, Comput. Methods Funct. Theory13(1) (2013), 133–151] proved that the similar set obtained for functions with half-integer coefficients consists of only 21 functions; that is, 12 more functions in addition to these nine functions of Friedman from the set ${\mathcal{S}}_{\mathbb{Z}}$. In this paper, we determine the class of all normalized sense-preserving univalent harmonic mappings $f$ on the unit disk with half-integer coefficients for the analytic and co-analytic parts of $f$. It is surprising to see that there are only 27 functions out of which only six functions in this class are not conformal. This settles the recent conjecture of the authors. We also prove a general result, which leads to a new conjecture.

Keywords

typically real analytic mappingsharmonic mappingsunivalentstarlikeconvexclose-to-convextypically real harmonic mappingsconvex in real directionconvex in imaginary directionSchwarz lemmasubordinationhalf-integer coefficientsuniformly discrete set

MSC classification

Primary: 30C65: Quasiconformal mappings in $^n$, other generalizations
Secondary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C20: Conformal mappings of special domains
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 98 , Issue 2 , April 2015 , pp. 257 - 280
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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