Advertisement

Computational Methods and Function Theory

, Volume 19, Issue 4, pp 583–599 | Cite as

Convolutions of Normalized Harmonic Mappings

  • Stacey MuirEmail author
Article
  • 78 Downloads

Abstract

Recent results on the convolution of two planar harmonic mappings is built upon the theory that when the convolution of functions from certain families of mappings, such as half-plane or strip mappings, is locally univalent, then the convolution will possess certain direction-convexity properties. Thus, much of the latest work on harmonic convolutions centers around establishing conditions on the dilatations of \(f_1, f_2: {\mathbb D}\rightarrow {\mathbb C}\) from the families above so that \(f_1 * f_2\) is locally univalent. Recently, it was noted that normalizations for these families were not treated properly when some dilatations considered did not fix zero. In this paper, we account for a variety of dilatations that do not fix zero by broadening the family from which \(f_1\) and \(f_2\) are chosen. Additionally, we show that when removing a hypothesis from one of our results it is possible to have a locally univalent convolution that fails to be univalent. This demonstrates that some of the previous work on convolutions cannot simply be modified by a re-normalization while affirming the necessity of the hypothesis.

Keywords

Harmonic mappings Convolution Univalence Convex in one direction 

Mathematics Subject Classification

30C45 

Notes

References

  1. 1.
    Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Annales Acad. Sci. Fenn. 9, 3–25 (1984)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dorff, M.: Convolutions of planar convex harmonic mappings. Complex Var. Theory Appl. 45, 263–271 (2001)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Dorff, M., Nowak, M., Woloszkiewicz, M.: Convolutions of harmonic convex mappings. Complex Var. Elliptic Equ. 57(5), 489–503 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Jiang, Y., Rasila, A., Sun, Y.: A note on convexity of convolutions of harmonic mappings. Bull. Korean Math. Soc. 52(6), 1925–1935 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kumar, R., Dorff, M., Gupta, S., Singh, S.: Convolution properties of some harmonic mappings in the right half-plane. Bull. Malays. Math. Sci. Soc. 39(1), 439–455 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Lewy, H.: On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull. Am. Math. Soc. 42, 689–692 (1936)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Li, L., Ponnusamy, S.: Solution to an open problem on convolutions of harmonic mappings. Complex Var. Elliptic Equ. 58(12), 1647–1653 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Liu, Z.H., Ponnusamy, S.: Univalency of convolutions of univalent harmonic right half-plane mappings. Comput. Methods Funct. Theory 17, 289–302 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Muir, S.: Convex combinations of planar harmonic mappings realized through convolutions with half-strip mappings. Bull. Malays. Math. Sci. Soc. 40(2), 857–880 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Royster, W.C., Ziegler, M.: Univalent functions convex in one direction. Publ. Math. Debrecen 23, 339–345 (1976)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe University of ScrantonScrantonUSA

Personalised recommendations