Monatshefte für Mathematik

, Volume 180, Issue 3, pp 527–548 | Cite as

Radii of covering disks for locally univalent harmonic mappings

  • Sergey Yu. Graf
  • Saminathan Ponnusamy
  • Victor V. Starkov
Article
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Abstract

For a univalent smooth mapping f of the unit disk \({\mathbb D}\) of complex plane onto the manifold \(f({\mathbb D})\), let \(d_f(z_0)\) be the radius of the largest univalent disk on the manifold \(f({\mathbb D})\) centered at \(f(z_0)\) (\(|z_0|<1\)). The main aim of the present article is to investigate how the radius \(d_h(z_0)\) varies when the analytic function h is replaced by a sense-preserving harmonic function \(f=h+\overline{g}\). The main result includes sharp upper and lower bounds for the quotient \(d_f(z_0)/d_h(z_0)\), especially, for a family of locally univalent Q-quasiconformal harmonic mappings \(f=h+\overline{g}\) on \(|z|<1\). In addition, estimate on the radius of the disk of convexity of functions belonging to certain linear invariant families of locally univalent Q-quasiconformal harmonic mappings of order \(\alpha \) is obtained.

Keywords

Locally univalent harmonic mappings Linear and affine invariant families Convex and close-to-convex functions Covering theorems 

Mathematics Subject Classification

30C62 31A05 30C45 
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Notes

Acknowledgments

The authors thank the referee for useful comments. The research was supported by the project RUS/RFBR/P-163 under Department of Science and Technology (India). The second author is currently on leave from IIT Madras. The third author is also supported by Russian Foundation for Basic Research (Project 14-01-00510) and the Strategic Development Program of Petrozavodsk State University.

References

  1. 1.
    Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997)MATHGoogle Scholar
  2. 2.
    Bieberbach, L.: Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. Sitzungsber. Preuss. Akad. Wiss. 38, 940–955 (1916)MATHGoogle Scholar
  3. 3.
    Chen, H., Gauthier, P.M., Hengartner, W.: Bloch constants for planar harmonic mappings. Proc. Am. Math. Soc. 128, 3231–3240 (2000)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chen, S., Ponnusamy, Rasila, A.: Coefficient estimates, Landau’s theorem and Lipschitz-type spaces on planar harmonic mappings. J. Aust. Math. Soc. 96, 198-215 (2014)Google Scholar
  5. 5.
    Chen, S., Ponnusamy, Wang, X.: Bloch and Landau’s theorems for planar \(p\)-harmonic mappings. J. Math. Anal. Appl. 373, 102-110 (2011)Google Scholar
  6. 6.
    Chen, Sh, Ponnusamy, S., Wang, X.: Landau’s theorem and Marden constant for harmonic \(\nu \)-Bloch mappings. Bull. Aust. Math. Soc. 84, 19–32 (2011)MathSciNetMATHGoogle Scholar
  7. 7.
    Chen, Sh, Ponnusamy, S., Wang, X.: Properties of some classes of planar harmonic and planar biharmonic mappings. Complex Anal. Oper. Theory 5, 901–916 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen, Sh, Ponnusamy, S., Wang, X.: Coefficient estimates and Landau–Bloch’s theorem for planar harmonic mappings. Bull. Malays. Math. Sci. Soc. 34(2), 255–265 (2011)MathSciNetMATHGoogle Scholar
  9. 9.
    Chen, Sh, Ponnusamy, S., Wang, X.: Covering and distortion theorems for planar harmonic univalent mappings. Arch. Math. (Basel) 101, 285–291 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chen, Sh, Ponnusamy, S., Wang, X.: Harmonic mappings in Bergman spaces. Monatsh. Math. 170, 325–342 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Clunie, J.G., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fenn. Ser. A I Math. 9, 3-25 (1984)Google Scholar
  12. 12.
    Duren, P.L.: Univalent Functions, Grundlehren der mathematischen Wissenschaften, vol. 259. Springer, New York (1983)Google Scholar
  13. 13.
    Duren, P.: Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, vol. 156. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  14. 14.
    Goluzin, G.M.: Geometric Theory of Functions of a Complex Variable, Translations of Mathematical Monographs, vol. 26. American Mathematical Society, Providence (1969)Google Scholar
  15. 15.
    Graf, S.Yu., Eyelangoli, O.R.: Differential inequalities in linear and affine-invariant families of harmonic mappings. Russian Math. (Iz. VUZ) 54(10), 60-62 (2010)Google Scholar
  16. 16.
    Koebe, P.: Über die Uniformisierung beliebiger analytischer Kurven II, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Math.-Phys. Kl., 633-669 (1907)Google Scholar
  17. 17.
    Lewy, H.: On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull. Am. Math. Soc. 42, 689–692 (1936)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Minda, D.: The Bloch and Marden Constants, Computational Methods and Function Theory (Valparaso, 1989), pp. 131-142, Lecture Notes in Math., vol. 1435. Springer, Berlin (1990)Google Scholar
  19. 19.
    Mori, A.: On an absolute constant in the theory of quasi-conformal mappings. J. Math. Soc. Japan 8, 156–166 (1956)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Pommerenke, Ch.: Linear-invariante familien analytischer funktionen. I. Math. Ann. 155, 108–154 (1964)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Pommerenke, Ch.: Univalent Functions. Vandenhoeck and Ruprecht, Göttingen (1975)MATHGoogle Scholar
  22. 22.
    Ponnusamy, S., Rasila, A.: Planar harmonic and quasiregular mappings. In: Ruschewyeh, St., Ponnusamy, S. (eds.) Topics in Modern Function Theory, Chapter in CMFT, RMS-Lecture Notes Series No., vol. 19, pp. 267-333 (2013)Google Scholar
  23. 23.
    Schaubroeck, L.E.: Subordination of planar harmonic functions. Complex Var. 41, 163–178 (2000)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Sheil-Small, T.: Constants for planar harmonic mappings. J. Lond. Math. Soc. 42, 237–248 (1990)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Sobczak-Knec, M., Starkov, V.V., Szynal, J.: Old and new order of linear invariant family of harmonic mappings and the bound for Jacobian. Ann. Univ. Mariae Curie-Skłodowska. Sectio A. Math. V. LXV(2), 191-202 (2011)Google Scholar
  26. 26.
    Starkov, V.V.: Equivalent definitions of linear invariant families, Materialy XI Konf. Szkolenovej z Teorii Zagadnien Ekstremalnych, Lodz., pp. 34-38 (1990, in Polish)Google Scholar
  27. 27.
    Starkov, V.V.: Harmonic locally quasiconformal mappings. Tr. Petrozavodsk Gos. Univ. Ser. Mat. 1, 61–69 (1993). (in Russian)MathSciNetMATHGoogle Scholar
  28. 28.
    Starkov, V.V.: Harmonic locally quasiconformal mappings. Ann. Univ. Mariae Curie-Skłodowska Sectio A. Math. 14, 183–197 (1995)MathSciNetMATHGoogle Scholar
  29. 29.
    Starkov, V.V.: Univalence disks of harmonic locally quasiconformal mappings and harmonic Bloch functions. Sib. Math. J. 38, 791–800 (1997)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  • Sergey Yu. Graf
    • 1
  • Saminathan Ponnusamy
    • 2
  • Victor V. Starkov
    • 3
  1. 1.Department of MathematicsTver State UniversityTverRussia
  2. 2.Indian Statistical Institute (ISI)Chennai Centre SETS (Society for Electronic Transactions and Security)ChennaiIndia
  3. 3.Department of MathematicsUniversity of PetrozavodskPetrozavodskRussia

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