Show Summary Details
More options …

# Analysis

### International mathematical journal of analysis and its applications

CiteScore 2018: 0.72

SCImago Journal Rank (SJR) 2018: 0.363
Source Normalized Impact per Paper (SNIP) 2018: 0.530

Mathematical Citation Quotient (MCQ) 2018: 0.36

Online
ISSN
2196-6753
See all formats and pricing
More options …
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
• Email
• Other articles by this author:
/ Jinjing Qiao
Published Online: 2016-09-29 | DOI: https://doi.org/10.1515/anly-2014-1293

## Abstract

Let ${\mathcal{𝒮}}_{H}$ denote the usual class of all normalized functions $f=h+\overline{g}$ 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 ${\mathcal{𝒮}}_{H}$ 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 ${\mathcal{𝒮}}_{H}$ 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.

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

## References

• [1]

Bharanedhar S. V. and Ponnusamy S., Coefficient conditions for harmonic univalent mappings and hypergeometric mappings, Rocky Mountain J. Math. 44 (2014), no. 3, 753–777. Google Scholar

• [2]

Bshouty D. and Lyzzaik A., Close-to-convexity criteria for planar harmonic mappings, Complex Anal. Oper. Theory 5 (2011), no. 3, 767–774. Google Scholar

• [3]

Clunie J. G. and Sheil-Small T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 9 (1984), 3–25. Google Scholar

• [4]

Dorff M., Convolutions of planar harmonic convex mappings, Complex Variables Theory Appl. 45 (2001), 263–271. Google Scholar

• [5]

Dorff M., Nowak M. and Wołoszkiewicz M., Convolutions of harmonic convex mappings, Complex Var. Elliptic Equ. 57 (2012), no. 5, 489–503. Google Scholar

• [6]

Duren P., Univalent Functions, Springer, New York, 1982. Google Scholar

• [7]

Duren P., Harmonic Mappings in the Plane, Cambridge University Press, Cambridge, 2004. Google Scholar

• [8]

Friedman B., Two theorems on schlicht functions, Duke Math. J. 13 (1946), 171–177. Google Scholar

• [9]

Goodman A. W., Univalent Functions. Vols. 1–2, Mariner, Tampa, 1983. Google Scholar

• [10]

Greiner P., Geometric properties of harmonic shears, Comput. Methods Funct. Theory 4 (2004), no. 1, 77–96. Google Scholar

• [11]

Gronwall T. H., Some remarks on conformal representation, Ann. of Math. (2) 16 (1914), 72–76. Google Scholar

• [12]

Hengartner W. and Schober G., On schlicht mappings to domains convex in one direction, Comment. Math. Helv. 45 (1970), 303–314. Google Scholar

• [13]

Hiranuma N. and Sugawa T., Univalent functions with half-integral coefficients, Comput. Methods Funct. Theory 13 (2013), no. 1, 133–151. Google Scholar

• [14]

Jenkins J. A., On univalent functions with integral coefficients, Complex Variables Theory Appl. 9 (1987), 221–226. Google Scholar

• [15]

Lecko A., On the class of functions convex in the negative direction of the imaginary axis, J. Aust. Math. Soc. 73 (2002), 1–10. Google Scholar

• [16]

Lewy H., On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689–692. Google Scholar

• [17]

Li L. and Ponnusamy S., Convolutions of slanted half-plane harmonic mappings, Analysis (Munich) 33 (2013), 1001–1018. Google Scholar

• [18]

Li L. and Ponnusamy S., Solution to an open problem on convolutions of harmonic mappings, Complex Var. Elliptic Equ. 58 (2013), no. 12, 1647–1653. Google Scholar

• [19]

Linis V., Note on univalent functions, Amer. Math. Monthly 62 (1955), 109–110. Google Scholar

• [20]

Obradović M. and Ponnusamy S., New criteria and distortion theorems for univalent functions, Complex Variables Theory Appl. 44 (2001), 173–191. Google Scholar

• [21]

Pommerenke C., Univalent Functions, Vandenhoeck and Ruprecht, Göttingen, 1975. Google Scholar

• [22]

Ponnusmay S. and Qiao J., Classification of univalent harmonic mappings on the unit disk with half-integer coefficients, J. Aust. Math. Soc. 98 (2015), 257–280. Google Scholar

• [23]

Ponnusamy S. and Rasila A., Planar harmonic and quasiregular mappings, Topics in Modern Function Theory (Guwahati 2008), Ramanujan Math. Soc. Lect. Notes Ser. 19, Ramanujan Mathematical Society, Mysore (2013), 267–333. Google Scholar

• [24]

Robertson M. S., Analytic functions starlike in one direction, Amer. J. Math. 58 (1936), 465–472. Google Scholar

• [25]

Rogosinski W. W., On the coefficients of subordinate functions, Proc. Lond. Math. Soc. (2) 48 (1943), 48–82. Google Scholar

• [26]

Royster W. C., Rational univalent functions, Amer. Math. Monthly 63 (1956), 326–328. Google Scholar

• [27]

Royster W. C. and Ziegler M., Univalent functions convex in one direction, Publ. Math. Debrecen 23 (1976), 339–345. Google Scholar

• [28]

Schaubroeck L. E., Growth, distortion and coefficient bounds for plane harmonic mappings convex in one direction, Rocky Mountain J. Math. 31 (2001), no. 2, 624–639. Google Scholar

• [29]

Shah T.-S., On the coefficients of Schlicht functions, J. Chinese Math. Soc. (N. S.) 1 (1951), 98–107. Google Scholar

• [30]

Townes S. B., A theorem on schlicht functions, Proc. Amer. Math. Soc. 5 (1954), 585–588. Google Scholar

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,

Export Citation