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Journal of Mathematical Sciences

, Volume 207, Issue 6, pp 825–831 | Cite as

Bounded Holomorphic Functions Covering No Concentric Circles

  • V. N. Dubinin
Article

Growth and distortion theorems for the functions indicated in the title are proved by the symmetrization method. Sharp estimates for the moduli of the functions considered and their derivatives at inner and boundary points are established, and an estimate for the Schwarzian derivative is obtained. Bibliography: 14 titles

Keywords

Boundary Point Sharp Estimate Schwarzian Derivative Arbitrary Real Number Distortion Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    W. K. Hayman, Multivalent Functions, Second ed., Cambridge Univ. Press, Cambridge (1994).CrossRefzbMATHGoogle Scholar
  2. 2.
    A. Frolova, M. Levenshtein, D. Shoikhet, and A. Vasil’ev, “ Boundary distortion estimates for holomorphic maps,” Complex Anal. Oper. Theory, 8, 1129–1149 (2014).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    V. Bolotnikov, M. Elin, and D. Shoikhet, “Inequalities for angular derivatives and boundary interpolation,” Anal. Math. Phys., 3, 63–96 (2013).CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    T. Aliyev Azerŏglu and B. N. Örnek, “A refined Schwarz inequality on the boundary,” Complex Var. Elliptic Eq., 58, 571–577 (2013).CrossRefzbMATHGoogle Scholar
  5. 5.
    A. Lecko and B. Uzar, “A note on Julia–Carathéodory theorem for functions with fixed initial coefficients,” Proc. Japan Acad., Ser. A, 89, 133–137 (2013).CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    B. N. Örnek, “Sharpened forms of the Schwarz lemma on the boudary,” Bull. Korean Math. Soc., 50, 2053-2059 (2013).CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    G. Cleanthous, “Growth theorems for holomorphic functions under geometric conditions for the image,” Comput. Meth. Funct. Theory, 13, 277–294 (2013).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    V. N. Dubinin, Condenser Capacities and Symmetrization in Geometric Function Theory [in Russian], Dal’nauka, Vladivostok (2009).Google Scholar
  9. 9.
    S. Pouliasis, “Condenser capacity and meromorphic functions,” Comput. Meth. Funct. Theory, 11, 237–245 (2011).CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    V. N. Dubinin, “On the preservation of conformal capacity under meromorphic functions,” Zap. Nauchn. Semin. POMI, 392, 67–73 (2011).Google Scholar
  11. 11.
    Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer (1992).Google Scholar
  12. 12.
    I. P. Mityuk, Application of Symmetrization Methods to Geometric Function Theory [in Russian], Kub. Gos. Univ., Krasnodar (1985).Google Scholar
  13. 13.
    V. N. Dubinin, “Schwarz lemma and estimates for the coefficients of regular functions with free domain of definition,” Mat. Sb., 196, No. 11, 53–74 (2005).CrossRefMathSciNetGoogle Scholar
  14. 14.
    V. N. Dubinin, “On boundary values of the Schwarzian derivative of a regular function,” Mat. Sb., 202, No. 5, 29–44 (2011).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Far Eastern Federal UniversityVladivostokRussia

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