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Lengths, Area and Modulus of Continuity of Some Classes of Complex-Valued Functions

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Abstract

In this paper, we discuss the modulus of continuity of solutions to Poisson’s equation, and give bounds of length and area distortion for some classes of K-quasiconformal mappings satisfying Poisson’s equations. The obtained results are the extension of the corresponding classical results.

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References

  1. Abaob, A., Arsenovic, M., Mateljević, M.: Moduli of continuity of harmonic quasiregular mappings on bounded domains. Ann. Acad. Sci. Fenn. Math. 38, 839–847 (2013)

    Article  MathSciNet  Google Scholar 

  2. Beckenbach, E.F.: A relative of the lemma of Schwarz. Bull. Am. Math. Soc. 44, 698–707 (1938)

    Article  MathSciNet  Google Scholar 

  3. Carleman, T.: Zur Theorie der Minimalflächen. Math. Z. 9, 154–160 (1921)

    Article  MathSciNet  Google Scholar 

  4. Chen, Sh, Liu, G., Ponnusamy, S.: Linear measure and K-quasiconformal harmonic mappings. Sci. Sin. Math. 47, 565–574 (2017). (in Chinese)

    Google Scholar 

  5. Chen, Sh, Ponnusamy, S., Rasila, A.: On characterizations of Bloch-type, Hardy-type and Lipschitz-type spaces. Math. Z. 279, 163–183 (2015)

    Article  MathSciNet  Google Scholar 

  6. Chen, Sh, Ponnusamy, S., Rasila, A.: Lengths, area and Lipschitz-type spaces of planar harmonic mappings. Nonlinear Anal. 115, 62–80 (2015)

    Article  MathSciNet  Google Scholar 

  7. Duren, P.: Theory of \(H^{p}\) Spaces, 2nd edn. Dover, Mineola (2000)

    Google Scholar 

  8. Dyakonov, K.M.: Holomorphic functions and quasiconformal mappings with smooth moduli. Adv. Math. 187, 146–172 (2004)

    Article  MathSciNet  Google Scholar 

  9. Dyakonov, K.M.: Equivalent norms on Lipschitz-type spaces of holomorphic functions. Acta Math. 178, 143–167 (1997)

    Article  MathSciNet  Google Scholar 

  10. Gehring, F.W., Martio, O.: Lipschitz-classes and quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I Math. 10, 203–219 (1985)

    Article  MathSciNet  Google Scholar 

  11. Holland, F., Walsh, D.: Criteria for membership of Bloch space and its subspace, \(BMOA\). Math. Ann. 273, 317–335 (1986)

    Article  MathSciNet  Google Scholar 

  12. Hörmander, L.: Notions of Convexity, Progress in Mathematics, vol. 127. Birkhäuser Boston Inc, Boston (1994)

    MATH  Google Scholar 

  13. Kalaj, D., Pavlović, M.: On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation. Trans. Am. Math. Soc. 363, 4043–4061 (2011)

    Article  MathSciNet  Google Scholar 

  14. Kalaj, D.: Cauchy transform and Poisson’s equation. Adv. Math. 231, 213–242 (2012)

    Article  MathSciNet  Google Scholar 

  15. Krantz, S.G.: Lipschitz spaces, smoothness of functions, and approximation theory. Expo. Math. 3, 193–260 (1983)

    MathSciNet  MATH  Google Scholar 

  16. Lappalainen, V.: Lip\(_{h}\)-extension domains. Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 56 (1985)

  17. Mateljević, M., Vuorinen, M.: On harmonic quasiconformal quasi-isometries. J. Inequal. Appl. 2010, Article ID 178732. https://doi.org/10.1155/2010/178732

    Article  MathSciNet  Google Scholar 

  18. Mateljević, M.: Distortion of quasiregular mappings and equivalent norms on Lipschitz-type spaces. Abstr. Appl. Anal. 2014, Article ID 895074. https://doi.org/10.1155/2014/895074

    Article  MathSciNet  Google Scholar 

  19. Pavlović, M.: On the Holland–Walsh characterization of Bloch functions. Proc. Edinb. Math. Soc. 51, 439–441 (2008)

    Article  MathSciNet  Google Scholar 

  20. Pavlović, M.: On Dyakonov’s paper equivalent norms on Lipschitz-type spaces of holomorphic functions. Acta Math. 183, 141–143 (1999)

    Article  MathSciNet  Google Scholar 

  21. Ren, G., Kähler, U.: Weighted Lipschitz continuity and harmonic Bloch and Besov spaces in the real unit ball. Proc. Edinb. Math. Soc. 48, 743–755 (2005)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank the referee for providing constructive comments and help in improving this paper. This research was partly supported by the Science and Technology Plan Project of Hengyang City (No. 2018KJ125), the Hunan Provincial Education Department Outstanding Youth Project (No. 18B365), the National Natural Science Foundation of China (No. 11571216), the Science and Technology Plan Project of Hunan Province (No. 2016TP1020), the Science and Technology Plan Project of Hengyang City (No. 2017KJ183), and the Application-Oriented Characterized Disciplines, Double First-Class University Project of Hunan Province (Xiangjiaotong [2018]469).

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  1. College of Mathematics and Statistics, Hengyang Normal University, Hengyang, 421008, Hunan, People’s Republic of China

    Shaolin Chen

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  1. Shaolin Chen
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Chen, S. Lengths, Area and Modulus of Continuity of Some Classes of Complex-Valued Functions. Results Math 74, 105 (2019). https://doi.org/10.1007/s00025-019-1028-5

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