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Uniformly locally univalent harmonic mappings

  • Saminathan Ponnusamy
  • Jinjing Qiao
  • Xiantao Wang
Article

Abstract

The primary aim of this paper is to characterize the uniformly locally univalent harmonic mappings in the unit disk. Then, we obtain sharp distortion, growth and covering theorems for one parameter family \({{\mathcal {B}}}_{H}(\lambda )\) of uniformly locally univalent harmonic mappings. Finally, we show that the subclass of k-quasiconformal harmonic mappings in \({{\mathcal {B}}}_{H}(\lambda )\) and the class \({{\mathcal {B}}}_{H}(\lambda )\) are contained in the Hardy space of a specific exponent depending on \(\lambda \), respectively, and we also discuss the growth of coefficients for harmonic mappings in \({{\mathcal {B}}}_{H}(\lambda )\).

Keywords

Harmonic mapping pre-Schwarzian derivatives uniformly locally univalence growth estimate coefficient estimate harmonic Bloch space Hardy space 

2010 Mathematics Subject Classification

Primary: 30C65 30C45 Secondary: 30C20 30C50 30C80 

Notes

Acknowledgements

The works of Mrs. Jinjing Qiao was supported by National Natural Science Foundation of China (No. 11501159), NSF of Hebei Science Foundation for Young Scientists (No. A2018201033) and was partially supported by INSA JRD-TATA Fellowship of the Centre for International Co-operation in Science (CICS). The third author (XW) was partly supported by NSFs of China (Nos. 11571216, 11671127 and 11720101003) and STU SRFT (No. 130-09400243).

References

  1. 1.
    Abu-Muhanna Y, Bloch, BMO and harmonic univalent functions, BMO and harmonic univalent functions, Complex Var. Theory Appl., 31 (1996) 271–279MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abu-Muhanna Y, Ali R M and Ponnusamy S, The spherical metric and harmonic univalent maps, Monatsh. Math., (2018) 14,  https://doi.org/10.1007/s00605-018-1160.4
  3. 3.
    Anderson J M, On Bloch functions and normal functions, J. Reine Angew. Math., 270 (1974) 12–37MathSciNetzbMATHGoogle Scholar
  4. 4.
    Becker J, Löwnersche Differentialgleichung and quasikonform fortsetzbare schlichte Funktionen, J. Reine Angew. Math., 255 (1972) 23–43MathSciNetzbMATHGoogle Scholar
  5. 5.
    Becker J and Pommerenke Ch., Schlichtheitskriterien und Jordangebriete, J. Reine Angew. Math., 354 (1984) 74–94MathSciNetzbMATHGoogle Scholar
  6. 6.
    Carleson L and Jones P W, On coefficient problems for univalent functions and conformal dimension, Duke Math. J., 66 (1992) 169–206MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen Sh., Ponnusamy S, Vuorinen M and Wang X, Lipschitz spaces and bounded mean oscillation of harmonic mappings, Bull. Aust. Math. Soc., 88(1) (2013) 143–157MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen Sh., Bloch constant and Landau’s theorem for planar \(p\)-harmonic mappings, J. Math. Anal. Appl., 373(1) (2011) 102–110MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen Sh., Landau–Bloch constants for functions in \(\alpha \)-Bloch spaces and Hardy spaces, Complex Anal. Oper. Theory, 6(5) (2012) 1025–1036MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Clunie J G and Sheil-Small T, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I., 9 (1984) 3–25MathSciNetzbMATHGoogle Scholar
  11. 11.
    Colonna F, The Bloch constant of bounded harmonic mappings, Indiana Univ. Math. J., 38 (1989) 829–840MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Colonna F, Bloch and normal functions and their relation, Rend. Circ. Mat. Palermo II, 38 (1989) 161–180MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Duren P, Harmonic mappings in the plane (2004) (New York: Cambridge University Press)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kim Y C, Some inequalities for uniformly locally univalent functions on the unit disk, Math. Inequal. Appl., 10 (2007) 805–809MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kim Y C and Sugawa T, Growth and coefficient estimates for uniformly locally univalent functions on the unit disk, Rocky Mt. J. Math., 32(1) (2002) 179–200MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kim Y C and Sugawa T, Uniformly locally univalent functions and Hardy spaces, J. Math. Anal. Appl., 353 (2009) 61–67MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lewy H, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Am. Math. Soc., 42 (1936) 689–692MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Makarov N G and Pommerenke Ch. (1997) On coefficients, boundary size and Hölder domains. Ann. Acad. Sci. Fenn. Ser. A I Math., 22, 305–312zbMATHGoogle Scholar
  19. 19.
    Noshiro K, On the star-shaped mapping by an analytic function, Proc. Imp. Acad. Jpn., 8 (1932) 275–277MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pommerenke Ch., Uniformly perfect sets and the Poincaré metric, Arch. Math. (Basel), 32(2) (1979) 192–199MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pommerenke Ch., Boundary behaviour of conformal maps (1992) (Berlin: Springer)CrossRefzbMATHGoogle Scholar
  22. 22.
    Ponnusamy S and Rasila A, Planar harmonic and quasiregular mappings, Topics in Modern Function Theory: Chapter in CMFT, RMS-Lecture Notes Series No. 19 (2013) pp. 267–333Google Scholar
  23. 23.
    Sugawa T, Various domain constants related to uniform perfectness, Complex Var. Theory Appl., 36(4) (1998) 311–345MathSciNetzbMATHGoogle Scholar
  24. 24.
    Yamashita S, Almost locally univalent functions, Monatsh. Math., 81 (1976) 235–240MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  • Saminathan Ponnusamy
    • 1
  • Jinjing Qiao
    • 2
  • Xiantao Wang
    • 3
  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia
  2. 2.Department of MathematicsHebei UniversityBaodingPeople’s Republic of China
  3. 3.Department of MathematicsShantou UniversityShantouPeople’s Republic of China

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