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International mathematical journal of analysis and its applications

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Volume 37, Issue 1


Characterization of univalent harmonic mappings with integer or half-integer coefficients

Saminathan Ponnusamy
  • Indian Statistical Institute (ISI), Chennai Centre, SETS (Society for Electronic Transactions and Security), MGR Knowledge City, CIT Campus, Taramani, Chennai 600113, India
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/ Jinjing Qiao
Published Online: 2016-09-29 | DOI: https://doi.org/10.1515/anly-2014-1293


Let 𝒮H denote the usual class of all normalized functions f=h+g¯ harmonic and sense-preserving univalent on the unit disk |z|<1. In this article we show that the set, consisting of those mappings f from 𝒮H for which all Taylor coefficients of the analytic and co-analytic parts of f are integers, consists of only nine functions. The second aim is to discuss the set 𝒮H of those functions which have half-integer coefficients. More precisely, we determine the set of univalent harmonic mappings with half-integer coefficients which are convex in real direction or convex in imaginary direction. This work generalizes the recent paper of Hiranuma and Sugawa. One of the examples generated in this way helps to disprove a conjecture of Bharanedhar and Ponnusamy.

Keywords: Harmonic mappings; univalent; subordination; convex and starlike functions; integer coefficients; half-integer coefficients; convex in real direction; convex in imaginary direction

MSC 2010: 31A05; 31C05; 30C45; 30C20


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About the article

Received: 2014-11-08

Revised: 2015-08-14

Accepted: 2016-03-13

Published Online: 2016-09-29

Published in Print: 2017-02-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11501159

The work of Mrs. Jinjing Qiao was supported by the Centre for International Co-operation in Science (CICS) through the award of “INSA JRD-TATA Fellowship”, and by the National Natural Science Foundation of China (grant no. 11501159). The work was completed during her visit to the Department of Mathematics, IIT Madras, between April and June 2012.

Citation Information: Analysis, Volume 37, Issue 1, Pages 23–38, ISSN (Online) 2196-6753, ISSN (Print) 0174-4747, DOI: https://doi.org/10.1515/anly-2014-1293.

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