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Distortion theorems for univalent meromorphic functions on an annulus

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Abstract

We apply the capacity and symmetrization methods to distortion theorems for analytic functions in an annulus. We show that the classical Teichmüller estimate for the capacity of a doubly-connected domain yields a series of the already known and new inequalities for univalent functions. In particular, we supplement the results of Grötzsch, Duren, and Huckemann. Using the dissymmetrization of condensers we establish sharp estimates for local distortion and the distortion of level curves in n ≥ 2 symmetric directions. In terms of Robin functions we give an analog of the Nehari inequality: some general distortion theorem for several points taking into account the boundary behavior of the function and describing the cases of equalities. As a corollary we give analogs of some inequalities of Solynin, Pommerenke, and Vasil’ev that were obtained previously for univalent and bounded functions in a disk. We prove a distortion theorem that involves the Schwarzian derivatives at symmetric points on the unit circle.

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References

  1. Dubinin V. N., “Generalized condensers and the asymptotics of their capacities under degeneration of some plates,” J. Math. Sci. (New York), 129, No. 3, 3835–3842 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  2. Dubinin V. N., “Symmetrization in the geometric theory of functions of a complex variable,” Uspekhi Mat. Nauk, 49, No. 1, 3–76 (1994).

    MathSciNet  Google Scholar 

  3. Dubinin V. N. and Kostyuchenko E. V., “The Teichmüller extremal problem and distortion theorems in the theory of univalent functions,” Siberian Math. J., 40, No. 2, 258–261 (1999).

    MathSciNet  Google Scholar 

  4. Teichmüller O., “Untersuchungen über konforme und quasikonforme Abbildung,” Deutsch. Math., 3, 621–678 (1938).

    MATH  Google Scholar 

  5. Grötzsch H., “Über einige Extremalprobleme der konformer Abbildung,” Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl., 80, pp. 367–376 (1928).

    Google Scholar 

  6. Duren P. L., “Distortion in certain conformal mappings of an annulus,” Michigan Math. J., 10, 431–441 (1963).

    Article  MathSciNet  Google Scholar 

  7. Huckemann F., “Extremal elements in certain classes of conformal mappings of an annulus,” Acta Math., 118, No. 3–4, 193–221 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  8. Akhiezer N. I., Elements of the Theory of Elliptic Functions [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  9. Dubinin V. N. and Kostyuchenko E. V., “Extremal problems in the function theory associated with the n-fold symmetry,” J. Math. Sci. (New York), 118, No. 1, 4778–4794 (2003).

    Article  MathSciNet  Google Scholar 

  10. Dubinin V. N., “The asymptotics of module of a degenerating condenser and some of its applications,” J. Math. Sci. (New York), 95, No. 3, 2209–2220 (1999).

    Article  MathSciNet  Google Scholar 

  11. Dubinin V. N. and Eyrikh N. V., “Some applications of generalized condensers to analytic function theory,” J. Math. Sci. (New York), 133, No. 6, 1634–1647 (2006).

    Article  Google Scholar 

  12. Dubinin V. N. and Prilepkina E. G., “Preservation of the generalized reduced module under some geometric transformations of the plane domains,” Dal’nevostochn. Mat. Zh., 6, No. 1–2, 39–56 (2005).

    Google Scholar 

  13. Kovalev L. V., “Monotonicity of generalized reduced module,” J. Math. Sci. (New York), 118, No. 1, 4861–4870 (2003).

    Article  MathSciNet  Google Scholar 

  14. Dubinin V. N. and Vuorinen M., “Robin functions and distortion theorems for regular mappings,” Math. Nachr. (to appear).

  15. Dubinin V. N., “Some covering theorems on the lines under a conformal mapping of the ring,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 9, 42–50 (1987).

  16. Nehari Z., “Some inequalities in the theory of functions,” Trans. Amer. Math. Soc., 75, No. 2, 256–286 (1953).

    Article  MATH  MathSciNet  Google Scholar 

  17. Duren P., “Robin capacity,” in: Computational Methods and Function Theory (CMFT’97), N. Papamichael, St. Ruscheweyh, E. B. Saff (Eds.), World Sci. Publ. Co., Singapore, 1999, pp. 177–190.

    Google Scholar 

  18. Henrici P., Applied and Computational Complex Analysis. Vol. 3, Wiley-Interscience, New York (1986).

    Google Scholar 

  19. Solynin A. Yu., “Boundary distortion and extremal problems in some classes of univalent functions,” Zap. Nauchn. Sem. POMI, 204, 115–142 (1993).

    Google Scholar 

  20. Pommerenke Ch. and Vasil’ev A., “On bounded univalent functions and the angular derivative,” Ann. Univ. Mariae Curie-Skłlodowska Sect. A, 8, 79–106 (2000).

    MathSciNet  Google Scholar 

  21. Pommerenke Ch. and Vasil’ev A., “Angular derivatives of bounded univalent functions and extremal partitions of the unit disk,” Pacific J. Math., 206, No. 2, 425–450 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  22. Markushevich A. I., The Theory of Analytic Functions. Vol. 1 [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  23. Antonyuk G. K., Some Problems of the Theory of Extremal Metrics [in Russian], Kubansk. Univ., Krasnodar (1983).

    Google Scholar 

  24. Jenkins J. A., “Some theorems on boundary distortion,” Trans. Amer. Math. Soc., 81, No. 2, 477–500 (1956).

    Article  MATH  MathSciNet  Google Scholar 

  25. Singh V., “Grunsky inequalities and coefficients of bounded schlicht functions,” Ann. Acad. Sci. Fenn. Ser. AI Math., No. 310, 1–21 (1962).

  26. Schiffer M., “Some new results in the theory of conformal mappings,” in: R. Courant, Dirichlet’s Principle, Conformal Mapping and Minimal Surfaces, Springer-Verlag, New York, Heidelberg, and Berlin (1977); Appendix, pp. 234–301.

    Google Scholar 

  27. Alenitsyn Yu. E., “On univalent functions without common values in the multiconnected domain,” Tr. Mat. Inst. Steklova, 94, 4–18 (1968).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics, Vladivostok, Russia

    V. N. Dubinin & E. G. Prilepkina

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  1. V. N. Dubinin
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  2. E. G. Prilepkina
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Correspondence to V. N. Dubinin.

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Original Russian Text Copyright © 2010 Dubinin V. N. and Prilepkina E. G.

The authors were supported by the Russian Foundation for Basic Research (Grant 08-01-00028), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2810.2008.1), and the Far East Division of the Russian Academy of Sciences (Grant 09-II-CO-01-003).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 2, pp. 285–302, March–April, 2010.

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Dubinin, V.N., Prilepkina, E.G. Distortion theorems for univalent meromorphic functions on an annulus. Sib Math J 51, 229–243 (2010). https://doi.org/10.1007/s11202-010-0023-6

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  • DOI: https://doi.org/10.1007/s11202-010-0023-6

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