A proof of a conjecture on convolution of harmonic mappings and some related problems

  • S. Yalçın Bursa Uludag Univ., Turkey
  • A. Ebadian Urmia Univ., Iran
  • S. Azizi Payame Noor Univ., Tehran, Iran
Keywords: Harmonic convolutions, harmonic vertical strip mappings, harmonic half-plane mappings

Abstract

UDC 517.5

Recently, Kumar et al. proposed a conjecture concerning the convolution of a generalized right half-plane mapping with a vertical strip mapping. They have verified the above conjecture for $n=1,2,3$ and $4$. Also, it has been proved only for $\beta=\pi/2$. In this paper, by using of a new method, we settle this conjecture in the affirmative for all $n\in\mathbb{N}$ and $\beta\in(0,\pi)$. Moreover, we will use this method to prove some results on convolution of harmonic mappings. This new method simplifies calculations and shortens the proof of results remarkably.

References

A. Cauchy, Exercises de mathematique, Oeuvres (2), 9 (1829).

J. Conway, Functions of one complex variable, Second Ed., Vol. I, Springer-Verlag (1978). DOI: https://doi.org/10.1007/978-1-4612-6313-5

M. Dorff, Anamorphosis, mapping problems, and harmonic univalent functions, Explorat. Complex Anal., 197 – 269 (2012). DOI: https://doi.org/10.1090/clrm/040/04

M. Dorff, Harmonic univalent mappings onto asymmetric vertical strips, Comput. Methods and Funct. Theory, 171 – 175 (1997).

M. Dorff, Convolution of planar harmonic convex mappings, Complex Var. Theory and Appl., 45, № 3, 263 – 271 (2001), https://doi.org/10.1080/17476930108815381 DOI: https://doi.org/10.1080/17476930108815381

M. Dorff, M. Nowak, M. Wołoszkiewicz, Convolution of harmonic convex mappings, Complex Var. Elliptic Equat., 57, № 5, 489 – 503 (2012), https://doi.org/10.1080/17476933.2010.487211 DOI: https://doi.org/10.1080/17476933.2010.487211

P. Duren, Harmonic mappings in the plane, Cambridge Tracts Math., 156, Cambridge Univ. Press, Cambridge (2004), https://doi.org/10.1017/CBO9780511546600 DOI: https://doi.org/10.1017/CBO9780511546600

R. B. Gardner, N. K. Govil, Enestrom – Kakeya theorem and some of its generalizations, Current Topics in Pure and Comput. Complex Anal., 171 – 199 (2014). DOI: https://doi.org/10.1007/978-81-322-2113-5_8

R. Kumar, M. Dorff, S. Gupta, S. Singh, Convolution properties of some harmonic mappings in the right half-plane, Bull. Malays. Math. Sci. Soc., 39, № 1, 439 – 455 (2016), https://doi.org/10.1007/s40840-015-0184-3 DOI: https://doi.org/10.1007/s40840-015-0184-3

R. Kumar, S. Gupta, S. Singh, M. Dorff, An application of Cohn’s rule to convolutions of univalent harmonic mappings, Rocky Mountain J. Math., 46, № 2, 559 – 570 (2016), https://doi.org/10.1216/RMJ-2016-46-2-559 DOI: https://doi.org/10.1216/RMJ-2016-46-2-559

R. Kumar, S. Gupta, S. Singh, M. Dorff, On harmonic convolutions involving a vertical strip mapping, Bull. Korean Math. Soc., 52, № 1, 105 – 123 (2015), https://doi.org/10.4134/BKMS.2015.52.1.105 DOI: https://doi.org/10.4134/BKMS.2015.52.1.105

H. Lewy, On the nonvanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc., 42, 689 – 692 (1936), https://doi.org/10.1090/S0002-9904-1936-06397-4 DOI: https://doi.org/10.1090/S0002-9904-1936-06397-4

L. Li, S. Ponnusamy, Convolutions of slanted half-plane harmonic mappings, Analysis (Munich), 33, № 2, 159 – 176 (2013), https://doi.org/10.1524/anly.2013.1170 DOI: https://doi.org/10.1524/anly.2013.1170

L. Li, S. Ponnusamy, Solution to an open problem on convolution of harmonic mappings, Complex Var. Elliptic Equat., 58, № 12, 1647 – 1653 (2013), https://doi.org/10.1080/17476933.2012.702111 DOI: https://doi.org/10.1080/17476933.2012.702111

Y. Li, Z. Liu, Convolution of harmonic right half-plane mappings, Open Math., 14, 789 – 800 (2016), https://doi.org/10.1515/math-2016-0069 DOI: https://doi.org/10.1515/math-2016-0069

Z. Liu, Y. Jiang, Y. Sun, Convolutions of harmonic half-plane mappings with harmonic vertical strip mappings, Filomat, 31, № 7, 1843 – 1856 (2017), https://doi.org/10.2298/FIL1707843L DOI: https://doi.org/10.2298/FIL1707843L

S. Muir, Weak subordination for convex univalent harmonic functions, J. Math. Anal. and Appl., 348, 689 – 692 (2008), https://doi.org/10.1016/j.jmaa.2008.08.015 DOI: https://doi.org/10.1016/j.jmaa.2008.08.015

S. Ponnusamy, A. Rasila, Planar harmonic and quasiregular mappings, Topics in Modern Function Theory, Chapter in CMFT, RMS-Lect. Notes Ser., № 19, 267 – 333 (2013).

Published
22.02.2021
How to Cite
Yalçın, S., A. Ebadian, and S. Azizi. “A Proof of a Conjecture on Convolution of Harmonic Mappings and Some Related Problems”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 2, Feb. 2021, pp. 283 -88, doi:10.37863/umzh.v73i2.94.
Section
Short communications