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NOTE ON THE CONVOLUTION OF HARMONIC MAPPINGS

Part of: Geometric function theory

Published online by Cambridge University Press:  13 February 2019

LIULAN LI  and
SAMINATHAN PONNUSAMY Open the ORCID record for SAMINATHAN PONNUSAMY [Opens in a new window]
LIULAN LI
Affiliation:
College of Mathematics and Statistics, Hunan Provincial Key Laboratory of Intelligent Information Processing and Application, Hengyang Normal University, Hengyang, Hunan 421002, PR China email lanlimail2012@sina.cn
SAMINATHAN PONNUSAMY*
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India email samy@iitm.ac.in
*
✉samy@iitm.ac.in
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Abstract

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Dorff et al. [‘Convolutions of harmonic convex mappings’, Complex Var. Elliptic Equ. 57(5) (2012), 489–503] formulated a question concerning the convolution of two right half-plane mappings, where the normalisation of the functions was considered incorrectly. In this paper, we reformulate the problem correctly and provide a solution to it in a more general form. We also obtain two new theorems which correct and improve related results.

Keywords

harmonicunivalentslanted half-plane mappingconvex mappingconvex in a directionconvolution

MSC classification

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Secondary: 30C20: Conformal mappings of special domains 30C62: Quasiconformal mappings in the plane
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 3 , June 2019 , pp. 421 - 431
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Footnotes

The work of the first author is supported by NSF of China (No. 11571216), Application-Oriented Characterized Disciplines, Double First-Class University Project of Hunan Province (Xiangjiaotong [2018] 469), the Science and Technology Plan Project of Hunan Province (No. 2016TP1020) and the Science and Technology Plan Project of Hengyang City (2017KJ183). The work of the second author is supported by the Mathematical Research Impact Centric Support of the Department of Science and Technology, India (MTR/2017/000367).

References

Aleman, A. and Constantin, A., ‘Harmonic maps and ideal fluid flows’, Arch. Ration. Mech. Anal. 204 (2012), 479–513.Google Scholar
Axler, S., Bourdon, P. and Ramey, W., Harmonic Function Theory, 2nd edn, Graduate Texts in Mathematics, 137 (Springer, New York, 2001).Google Scholar
Bharanedhar, S. V. and Ponnusamy, S., ‘Coefficient conditions for harmonic univalent mappings and hypergeometric mappings’, Rocky Mountain J. Math. 44(3) (2014), 753–777.Google Scholar
Bshouty, D. and Lyzzaik, A., ‘Problems and conjectures in planar harmonic mappings’, in: Proc. ICM2010 Satellite Conf. Int. Workshop on Harmonic and Quasiconformal Mappings, IIT Madras, Aug. 09–17, 2010 (eds. Minda, D., Ponnusamy, S. and Shanmugalingam, N.) , J. Analysis 18 (2010), 69–81.Google Scholar
Clunie, J. G. and Sheil-Small, T., ‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A.I. 9 (1984), 3–25.Google Scholar
Constantin, O. and Martin, M. J., ‘A harmonic maps approach to fluid flows’, Math. Ann. 369 (2017), 1–16.Google Scholar
Dorff, M., ‘Convolutions of planar harmonic convex mappings’, Complex Var. Theory Appl. 45 (2001), 263–271.Google Scholar
Dorff, M., Nowak, M. and Wołoszkiewicz, M., ‘Convolutions of harmonic convex mappings’, Complex Var. Elliptic Equ. 57(5) (2012), 489–503.Google Scholar
Duren, P., Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, 156 (Cambridge University Press, Cambridge, 2004).Google Scholar
Kumar, R., Dorff, M., Gupta, S. and Singh, S., ‘Convolution properties of some harmonic mappings in the right half-plane’, Bull. Malays. Math. Sci. Soc. 39 (2016), 439–455.Google Scholar
Kumar, R., Gupta, S., Singh, S. and Dorff, M., ‘On harmonic convolution involving a vertical strip mapping’, Bull. Korean Math. Soc. 52(1) (2015), 105–123.Google Scholar
Kumar, R., Gupta, S., Singh, S. and Dorff, M., ‘An application of Cohn’s rule to convolutions of univalent harmonic mappings’, Rocky Mountain J. Math. 46(2) (2016), 559–569.Google Scholar
Li, L. and Ponnusamy, S., ‘Solution to an open problem on convolutions of harmonic mappings’, Complex Var. Elliptic Equ. 58(12) (2013), 1647–1653.Google Scholar
Li, L. and Ponnusamy, S., ‘Convolutions of slanted half-plane harmonic mappings’, Analysis (Munich) 33 (2013), 159–176.Google Scholar
Liu, Z. and Ponnusamy, S., ‘Univalency of convolutions of univalent harmonic right half-plane mappings’, Comput. Meth. Funct. Theor. 17(2) (2017), 289–302.Google Scholar
Ponnusamy, S. and Qiao, J., ‘Classification of univalent harmonic mappings on the unit disk with half-integer coefficients’, J. Aust. Math. Soc. 98(2) (2015), 257–280.Google Scholar
Ponnusamy, S. and Rasila, A., ‘Planar harmonic and quasiregular mappings’, in: Topics in Modern Function Theory, RMS Lecture Notes Series, 19 (eds. Ruscheweyh, St. and Ponnusamy, S.) (Ramanujan Mathematical Society, Mysore, India, 2013), 267–333.Google Scholar
Rahman, Q. T. and Schmeisser, G., Analytic Theory of Polynomials, London Mathematical Society Monographs, New Series, 26 (Oxford University Press, Oxford, 2002).Google Scholar
Ruscheweyh, St. and Sheil-Small, T., ‘Hadamard products of schlicht functions and the Pólya–Schoenberg conjecture’, Comment. Math. Helv. 48 (1973), 119–135.Google Scholar