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Counterexamples Concerning Quasiconformal Extensions of Strongly Starlike Functions

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Abstract

M. Fait, J. Krzyz and J. Zygmunt proved that a strongly starlike function of order α on the unit disk can be extended to a k-quasiconformal mapping with k ≤ sin(απ/2) on the whole complex plane \( \bar {\Bbb C} \) which fixes the point at infinity. An open question is whether such a function can be extended to a k-quasiconformal mapping with k ≤ α to the whole plane \( \bar {\Bbb C} .\)In this paper we will give a negative approach to the question.

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References

  1. Stankiewicz, J.: Quelques problémes extrémaux dans les classes des fonctions α-angulairement étoilées. Ann. Univ. Marie Curie–Sklodoώska Sect. A., XX, 59–75 (1966)

    MathSciNet  Google Scholar 

  2. Brannan, D. A., Kirwan, W. E.: On some classes of bounded univalent functions. J. London Math. Soc., 1, 431–443 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  3. Fait, M., Krzyz, J. G., Zygmunt, J.: Explicit quasiconformal extensions for some classes of univalent functions. Comment. Math. Helv., 51, 279–285 1976

    Article  MATH  MathSciNet  Google Scholar 

  4. Chiang, Y. M.: On a subclass of strongly gamma-starlike functions and quasiconformal extensions. Ann. Univ. Marie Curie–Sklodowska, Section A, 45, 21–27 (1991)

    MATH  MathSciNet  Google Scholar 

  5. Ma, W., Minda, D.: An internal geometric characterization of strongly starlike functions. Ann. Univ. Marie Curie–Sklodowska, Section A, 45, 89–97 (1991)

    MATH  Google Scholar 

  6. Sugawa, T.: On the norm of pre-Schwarzian derivatives of strongly starlike functions. Ann. Univ. Marie Curie–Sklodowska, Section A, 52, 149–157 (1998)

    MATH  MathSciNet  Google Scholar 

  7. Sugawa, T.: Holomorphic motions and quasiconformal extensions. Ann. Univ. Marie Curie–Sklodowska, Section A, 53, 239–252 (1998)

    MathSciNet  Google Scholar 

  8. Sugawa, T.: Inner radius of univalence for a strongly starlike domain. Monatsh. Math., 139, 61–68 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Sugawa, T.: A self-duality of strong starlikeness. Kodai Math. J., 28, 382–389 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Shen, Y.: A note on quasiconformal extensions of strongly starlike functions. Complex Variables, 49, 927–940 (2004)

    MATH  Google Scholar 

  11. Ahlfors, L. V.: Remarks on the Neumann–Poincare integral equation. Pacific J. Math., 2, 271–280 (1952)

    MATH  MathSciNet  Google Scholar 

  12. Pommerenke, Ch.: Univalent Functions, Vandenhoeck and Ruprecht, 1975

  13. Kühnau, R.: Wertannahmeprobleme bei quasikonformen Abbildungen mit ortsabhängiger Dilatationsbeschränkung. Math. Nachr, 40, 1–11 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lehto, O.: Schlicht functions with a quasiconformal extension. Ann. Acad. Sci. Fenn. A I Math., 500, 1–10 (1971)

    Google Scholar 

  15. Krushkal, S. L.: On the Grunsky coefficients. Siberian Math. J., 28, 138–145 (1987)

    Article  MathSciNet  Google Scholar 

  16. Krushkal, S. L.: Grunsky coefficient inequalities, Carathéodory metric and extremal quasiconformal mappings. Comment. Math. Helv., 64, 650–660 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  17. Shen, Y.: Pull-back operators by quasisymmetric functions and invariant metrics on Teichmüller spaces. Complex Variables, 42, 289–307 (2000)

    MATH  Google Scholar 

  18. Kühnau, R.: Zu den Grunskyschen Koeffizientenbedingungen. Ann. Acad. Sci. Fenn., 6, 125–130 (1981)

    MATH  Google Scholar 

  19. Schiffer, M.: Fredholm eigenvalues and Grunsky matrices. Ann. Pol. Math., 39, 149–164 (1981)

    MathSciNet  Google Scholar 

  20. Harmelin, R.: Bergman kernel function and univalent criteria. J. Anal. Math., 41, 249–258 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  21. Strebel, K.: Extremal quasiconformal mappings. Result. Math., 10, 169–209 (1986)

    MathSciNet  Google Scholar 

  22. Gardiner, F. P.: Teichmüller Theory and Quadratic Differentials, Wiley–Interscience, New York, 1987

  23. Fehlmann, R.: Quasiconformal mappings with free boundary components. Ann. Acad. Sci. Fenn., 7, 337–347 (1982)

    MathSciNet  MATH  Google Scholar 

  24. Strebel, K.: Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises. Comment. Math. Helv., 36, 306–323 (1962)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Suzhou University, Suzhou, 215006, P. R. China

    Yu Liang Shen

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  1. Yu Liang Shen
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Correspondence to Yu Liang Shen.

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This research was supported by NNSF of China (Grant No. 10231040) and NCET (06-0504)

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Shen, Y.L. Counterexamples Concerning Quasiconformal Extensions of Strongly Starlike Functions. Acta Math Sinica 23, 1859–1868 (2007). https://doi.org/10.1007/s10114-007-0954-4

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