Abstract and Applied Analysis

The Properties of a New Subclass of Harmonic Univalent Mappings

Zhi-Hong Liu and Ying-Chun Li

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Abstract

We introduced a new subclass of univalent harmonic functions defined by the shear construction in the present paper. First, we showed that the convolutions of two special subclass harmonic mappings are convex in the horizontal direction. Secondly, we proved a necessary and sufficient condition for the above subclass of harmonic mappings to be convex in the horizontal direction. We also presented some basic examples of univalent harmonic functions explaining the behavior of the image domains.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 794108, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393512083

Digital Object Identifier
doi:10.1155/2013/794108

Mathematical Reviews number (MathSciNet)
MR3116360

Zentralblatt MATH identifier
07095362

Citation

Liu, Zhi-Hong; Li, Ying-Chun. The Properties of a New Subclass of Harmonic Univalent Mappings. Abstr. Appl. Anal. 2013 (2013), Article ID 794108, 7 pages. doi:10.1155/2013/794108. https://projecteuclid.org/euclid.aaa/1393512083


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References

  • J. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae A , pp. 3–25, 1984.
  • P. L. Duren, Univalent functions, vol. 259, Springer, New York, NY, USA, 1983.
  • W. Hengartner and G. Schober, “On schlicht mappings to domains convex in one direction,” Commentarii Mathematici Helvetici, vol. 45, pp. 303–314, 1970.
  • M. Dorff, “Convolutions of planar harmonic convex mappings,” Complex Variables, Theory and Application, vol. 45, no. 3, pp. 263–271, 2001.
  • M. Dorff, M. Nowak, and M. Wołoszkiewicz, “Convolutions of harmonic convex mappings,” Complex Variables and Elliptic Equations, vol. 57, no. 5, pp. 489–503, 2012.
  • M. R. Goodloe, “Hadamard products of convex harmonic mappings,” Complex Variables, Theory and Application, vol. 47, no. 2, pp. 81–92, 2002.
  • L. L. Li and S. Ponnusamy, “Solution to an open problem on convolutions of harmonic mappings,” Complex Variables and Elliptic Equations, 2012.
  • L. L. Li and S. Ponnusamy, “Convolutions of slanted half-plane harmonic mappings,” Analysis, vol. 33, pp. 159–176, 2013.
  • Y. Abu-Muhanna and G. Schober, “Harmonic mappings onto convex domains,” Canadian Journal of Mathematics, vol. 39, no. 6, pp. 1489–1530, 1987.
  • W. Hengartner and G. Schober, “Univalent harmonic functions,” Transactions of the American Mathematical Society, vol. 299, no. 1, pp. 1–31, 1987.
  • C. Pommerenke, “On starlike and close-to-convex functions,” Proceedings of the London Mathematical Society, vol. 13, pp. 290–304, 1963.
  • J. S. Cheng, “A parallel algorithm for finding roots of a complex polynomial,” Journal of Computer Science and Technology, vol. 5, no. 1, pp. 71–81, 1990.