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Precise values of the Bloch constants of certain log-p-harmonic mappings

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Abstract

The aim of this article is twofold. One aim is to establish the precise forms of Landau-Bloch type theorems for certain polyharmonic mappings in the unit disk by applying a geometric method. The other is to obtain the precise values of Bloch constants for certain log-p-harmonic mappings. These results improve upon the corresponding results given in Bai et al. (Complex Anal. Oper. Theory, 13(2): 321–340, 2019).

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References

  1. Abdulhadi Z, Muhanna Y, Khuri S. On univalent solutions of the biharmonic equations. J Inequal Appl, 2005, 5: 469–478

    MathSciNet  MATH  Google Scholar 

  2. Abdulhadi Z, Muhanna Y. Landau’s theorems for biharmonic mappings. J Math Anal Appl, 2008, 338: 705–709

    Article  MathSciNet  Google Scholar 

  3. Bai X X, Liu M S. Landau-type theorems of polyharmonic mappings and log-p-harmonic mappings. Complex Anal Oper Theory, 2019, 13(2): 321–340

    Article  MathSciNet  Google Scholar 

  4. Chen H-H, Gauthier P M, Hengartner W. Bloch constants for planar harmonic mappings. Proc Amer Math Soc, 2000, 128(11): 3231–3240

    Article  MathSciNet  Google Scholar 

  5. Chen H-H, Gauthier P. The Landau theorem and Bloch theorem for planar Harmonic and pluriharmonic mappings. Proc Amer Math Soc, 2011, 139: 583–595

    Article  MathSciNet  Google Scholar 

  6. Chen S H, Ponnusamy S, Wang X. Landau’s theorems for certain biharmonic mappings. Appl Math Comput, 2009, 208(2): 427–433

    MathSciNet  MATH  Google Scholar 

  7. Chen S H, Ponnusamy S, Wang X. Bloch constant and Landau’s theorem for planar p-harmonic mappings. J Math Appl Appl, 2011, 373: 102–110

    Article  MathSciNet  Google Scholar 

  8. Chen S H, Ponnusamy S, Wang X. Properties of some classes of planar harmonic and planar biharmonic mappings. Complex Anal Oper Theory, 2011, 5: 901–916

    Article  MathSciNet  Google Scholar 

  9. Chen S H, Ponnusamy S, Rasila A. Coefficient estimates, Landau’s theorem and Lipschitz-type spaces on planar harmonic mappings. J Aust Math Soc, 2014, 96(2): 198–215

    Article  MathSciNet  Google Scholar 

  10. Chen S H, Ponnusamy S, Wang X. Coefficient estimates and Landau-Bloch’s constant for planar harmonic mappings. Bull Malaysian Math Sci Soc Second Series, 2011, 34(2): 255–265

    MathSciNet  MATH  Google Scholar 

  11. Chen J, Rasila A, Wang X. Landau’s theorem for polyharmonic mappings. J Math Appl Appl, 2014, 409: 934–945

    Article  MathSciNet  Google Scholar 

  12. Clunie J G, Sheil-Small T. Harmonic univalent functions. Ann Acad Sci Fenn, Ser AI, 1984, 9: 3–25

    MathSciNet  MATH  Google Scholar 

  13. Dorff M, Nowak M. Landau’s theorem for planar harmonic mappings. Comput Meth Funct Theory, 2004, 4(1): 151–158

    Article  MathSciNet  Google Scholar 

  14. Graham I, Kohr G. Geometric Function Theory in One and Higher Dimensions. New York: Marcel Dekker Inc, 2003

    Book  Google Scholar 

  15. Grigoryan A. Landau and Bloch theorems for harmonic mappings. Complex Variable Theory Appl, 2006, 51(1): 81–87

    Article  MathSciNet  Google Scholar 

  16. He L, Tu Z H. The schwarz lemma at the boundary of the non-convex complex ellipsoids. Acta Math Sci, 2019, 39B(4): 915–926

    Article  MathSciNet  Google Scholar 

  17. Huang X Z. Estimates on Bloch constants for planar harmonic mappings. J Math Anal Appl, 2008, 337: 880–887.

    Article  MathSciNet  Google Scholar 

  18. Huang X Z. Sharp estimate on univalent radius for planar harmonic mappings with bounded Frechet derivative (in Chinese). Sci Sin Math, 2014, 446: 685–692

    Article  Google Scholar 

  19. Lewy H. On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull Amer Math Soc, 1936, 42: 689–692

    Article  MathSciNet  Google Scholar 

  20. Li P, Ponnusamy S, Wang X. Some properties of planar p-harmonic and log-p-harmonic mappings. Bull Malays Math Sci Soc, 2013, 36(3): 595–609

    MathSciNet  MATH  Google Scholar 

  21. Li P, Wang X. Landau’s theorem for log-p-harmonic mappings. Appl Math Comput, 2012, 218: 4806–4812

    MathSciNet  MATH  Google Scholar 

  22. Liu M S. Landau’s theorems for biharmonic mappings. Complex Variables and Elliptic Equations, 2008, 53(9): 843–855

    Article  MathSciNet  Google Scholar 

  23. Liu M S. Estimates on Bloch constants for planar harmonic mappings. Sci China Ser A-Math, 2009, 52(1): 87–93

    Article  MathSciNet  Google Scholar 

  24. Liu M S. Landau’s theorem for planar harmonic mappings. Comput Math Appl, 2009, 57(7): 1142–1146

    Article  MathSciNet  Google Scholar 

  25. Liu M S, Chen H-H. The Landau-Bloch type theorems for planar harmonic mappings with bounded dilation. J Math Anal Appl, 2018, 468(2): 1066–1081

    Article  MathSciNet  Google Scholar 

  26. Liu M S, Luo L F, Luo X. Landau-Bloch type theorems for strongly bounded harmonic mappings. Monatshefte für Mathematik, 2020, 191(1): 175–185

    Article  MathSciNet  Google Scholar 

  27. Liu M S, Liu Z X. Landau-type theorems for p-harmonic mappings or log-p-harmonic mappings. Applied Analysis, 2014, 51(1): 81–87

    MathSciNet  MATH  Google Scholar 

  28. Liu M S, Wu F, Yang Y. Sharp estimates of quasi-convex mappings of type B and order alpha. Acta Math Sci, 2019, 39 B(5): 1265–1276

    Article  Google Scholar 

  29. Liu M S, Xie L, Yang L M. Landau’s theorems for biharmonic mappings (II). Mathematical Methods Applied Sciences, 2017, 40: 2582–2595

    Article  MathSciNet  Google Scholar 

  30. Liu M S, Luo L F. Landau-type theorems for certain bounded biharmonic mappings. Results in Mathematics, 2019, 74, Art 170

  31. Xia X Q, Huang X Z. Estimates on Bloch constants for planar bounded harmonic mappings (in Chinese). Chinese Annals of Mathematics A, 2010, 31(6): 769–776

    MathSciNet  MATH  Google Scholar 

  32. Zhu J F. Landau theorem for planar harmonic mappings. Complex Anal Oper Theory, 2015, 9: 1819–1826

    Article  MathSciNet  Google Scholar 

  33. Zhu Y C, Liu M S. Landau-type theorems for certain planar harmonic mappings or biharmonic mappings. Complex Variables and Elliptic Equations, 2013, 58(12): 1667–1676

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Mingsheng Liu & Lifang Luo

Authors
  1. Mingsheng Liu
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  2. Lifang Luo
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Corresponding author

Correspondence to Mingsheng Liu.

Additional information

This research was supported by Guangdong Natural Science Foundation (2018A030313508).

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Liu, M., Luo, L. Precise values of the Bloch constants of certain log-p-harmonic mappings. Acta Math Sci 41, 297–310 (2021). https://doi.org/10.1007/s10473-021-0117-1

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  • DOI: https://doi.org/10.1007/s10473-021-0117-1

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