Rocky Mountain Journal of Mathematics

An application of Cohn's rule to convolutions of univalent harmonic mappings

Raj Kumar, Sushma Gupta, Sukhjit Singh, and Michael Dorff

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Dorff et al.~\cite {do and no} proved that the harmonic convolutions of the standard right half-plane mapping $F_0=H_0+\overline {G}_0$ (where $H_0+G_0=z/(1-z)$ and $G_0'=-zH_0'$) and mappings $f_\beta =h_\beta +\overline {g}_{\beta }$ (where $f_\beta $ are obtained by shearing of analytic vertical strip mappings with dilatation $e^{i\theta }z^n$, $n=1,2$, $\theta \in \mathbb {R}$) are in $S_H^0$ and are convex in the direction of the real axis. In this paper, by using Cohn's rule, we generalize this result by replacing the standard right half-plane mapping $F_0$ with a family of right half-plane mappings $F_a=H_a+\overline {G}_a$ (with $H_a+G_a=z/(1-z)$ and $G'_a/H'_a= {(a-z)}/{(1-az)}$, $a\in (-1,1)$) and including the cases $n=3$ and $n=4$ (in addition to $n=1$ and $n=2$) for dilatations of $f_\beta $.

Article information

Rocky Mountain J. Math., Volume 46, Number 2 (2016), 559-570.

First available in Project Euclid: 26 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

Cohn's rule univalent harmonic mappings harmonic convolutions


Kumar, Raj; Gupta, Sushma; Singh, Sukhjit; Dorff, Michael. An application of Cohn's rule to convolutions of univalent harmonic mappings. Rocky Mountain J. Math. 46 (2016), no. 2, 559--570. doi:10.1216/RMJ-2016-46-2-559.

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  • J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Math. 9 (1984), 3–25.
  • M. Dorff, Convolutions of planar harmonic convex mappings, Comp. Var. Theor. Appl. 45 (2001), 263–271.
  • ––––, Harmonic mappings onto asymmetric vertical strips, in Computational methods and function theory, N. Papamichael, St. Ruscheweyh and E.B. Saff, eds., World Scientific Publishing, River Edge, NJ, 1999.
  • M. Dorff, M. Nowak and M. Woloszkiewicz, Convolutions of harmonic convex mappings, Comp. Var. Ellip. Eqn. 57 (2012), 489–503.
  • P.L. Duren, Harmonic mappings in the plane, Cambridge University Press, Cambridge, 2004.
  • M. Goodloe, Hadamard products of harmonic mappings, Comp. Var. Theor. Appl. 47 (2002), 81–92.
  • R. Kumar, M. Dorff, S.Gupta and S.Singh, Convolution properties of some harmonic mappings in the right half-plane, Bull. Malays. Math. Sci. Soc. 39 (2016), 439–455.
  • H. Lewy, On the non vanishing of the jacobian in certain one to one mappings, Bull. Amer. Math. Soc. 42 (1936), 689–692.
  • L. Li and S. Ponnusamy, Solution to an open problem on convolutions of harmonic mappings, Comp. Var. Ellip. Eqn. 58 (2013), 1647–1653.
  • ––––, Convolution of slanted right half-plane harmonic mappings, Analy. Int. J Anal. Appl. 32 (2013), 159–176.
  • Q.I. Rahman and G. Schmeisser, Analytic theory of polynomials, Lond. Math. Soc. Mono. 26, Oxford University Press, Oxford, 2002.
  • St. Ruscheweyh and T. Sheil-Small, Hadamard products of schlicht functions and the Polya-Schoenberg conjecture, Comment Math Helv. 48 (1973), 119–135.