Harmonic mappings and finite valency of analytic functions

L.A. Xuan

Kazan Federal University, Kazan, 420008 Russia

Can Tho University of Technology, Can Tho, 900000, Vietnam E-mail: laxuan@ctuet.edu.vn

Received August 15, 2018

Full text PDF

Abstract

The harmonic mappings of a circle into domains that can be cut into a finite number of convex subdomains has been studied. An estimate of the valency of holomorphic mappings of a circle composed of analytical components of the harmonic function has been obtained. In addition, the Keobe harmonic univalent function has been considered to illustrate the results of the research.

Keywords: harmonic function, univalent function, n-valent function, convex function, shear construction

Acknowledgments. We are grateful to Professor I.R. Kaymov, Doctor of Physics and Mathematics, for his valuable advice during the discussion of the results described in this paper.

References

  1. Bieberbach L. Uber die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbil-¨ dung des Einheitskreises vermitteln. S.-B. Preuss. Akad. Wiss. Berlin, 1916, pp. 940–955. (In German)
  2. Brilleslyper M.A., Dorff M.J., McDougall J.M., Rolf J.S., Schaubroeck L.E., Stankewitz R.L., Stephenson K. Explorations in Complex Analysis. Washington, Math. Assoc. Am., 2012. 425 p.
  3. Clunie J.G., Sheil-Small T. Harmonic univalent functions. Ann. Acad. Sci. Fenn. Ser. A. I., 1984, vol. 9, pp. 3–25.
  4. Duren P. Harmonic Mappings in the Plane. Cambridge, Cambridge Univ. Press, 2004. 226 p. doi: 10.1017/CBO9780511546600.
  5. Goodman A.W. An invitation to the study of univalent and multivalent functions. Int. J. Math. Math. Sci., 1979, vol. 2, no. 2, pp. 163–186. doi: 10.1155/S016117127900017X.
  6. Kayumov I.R., Ponnusamy S., Xuan L.A. Rotations of convex harmonic univalent mappings. Bull. Sci. Math., 2019. doi: 10.1016/j.bulsci.2019.01.007.
  7. Ponnusamy S., Kaliraj A.S. On the coefficient conjecture of Clunie and Sheil-Small on univalent harmonic mappings. Proc. Math. Sci., 2015, vol. 125, no. 3, pp. 277–290. doi: 10.1007/s12044-015-0236-5.

 

For citation: Xuan L.A. Harmonic mappings and finite valency of analytic functions. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 4, pp. 771–777. (In Russian)

 

The content is available under the license Creative Commons Attribution 4.0 License.