Advertisement

Journal of Mathematical Sciences

, Volume 207, Issue 6, pp 832–838 | Cite as

Inequalities for the Moduli of Circumferentially mean p-Valent Functions

  • V. N. Dubinin
Article

Let f be a circumferentially mean p-valent function in the disk |z| < 1 with Montel’s normalization f(0) = 0, f(ω) = ω (0 < ω < 1). Under an additional assumption on the covering of concentric circles by f, sharp lower and upper bounds on the modulus |f(z)| for some z ∈ (−1, 0) are established. It is shown that for nontrivial bounds to hold, such an assumption is necessary. Bibliography: 15 titles.

Keywords

Riemann Surface Branch Point Univalent Function Conformal Mapping Jordan Curve 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. Montel, Leçons sur les Fonctions Univalentes ou Multivalentes, Gauthier–Villars, Paris (1933).MATHGoogle Scholar
  2. 2.
    J. Krzyz, “On univalent functions with two preassigned values,” Ann. Univ. Mariae Curie-Sklodowska, Sec. A, 15, No. 5, 57–77 (1961).MATHMathSciNetGoogle Scholar
  3. 3.
    A. Vasil’ev, “Moduli of families of curves for conformal and quasiconformal mappings,” Lect. Notes Math., 1788, Springer (2002).Google Scholar
  4. 4.
    V. Singh, “Some extremal problems for a new class of univalent functions,” J. Math. Mech., 7, 811–821 (1958).MATHMathSciNetGoogle Scholar
  5. 5.
    Z. Lewandowski, “Sur certaines classes de fonctions univalentes dans le cercle unité,” Ann. Univ. Mariae Curie-Sklodowska, Sec. A, 13, No. 6, 115–126 (1959).MathSciNetGoogle Scholar
  6. 6.
    J. Krzyz and E. Zlotkiewicz, “Koebe sets for univalent functions with two preassigned values,” Ann. Acad. Sci. Fenn., Ser. A1. Math., 487 (1971).Google Scholar
  7. 7.
    R. J. Libera and E. J. Zlotkiewicz, “Bounded Montel univalent functions,” Colloq. Math., 56, 169–177 (1988).MATHMathSciNetGoogle Scholar
  8. 8.
    A. Vasil’ev and P. Pronin, “On some extremal problems for bounded univalent functions with Montel’s normalization,” Demonstratio Math., 26, 703–707 (1993).MathSciNetGoogle Scholar
  9. 9.
    A. Vasil’ev and P. Pronin, “The range of a system of functionals for the Montel univalent functions,” Bol. Soc. Mat. Mexicana, 6, 147–190 (2000).MathSciNetGoogle Scholar
  10. 10.
    W. K. Hayman, Multivalent functions, 2nd ed., Cambridge Univ. Press, Cambridge (1994).CrossRefMATHGoogle Scholar
  11. 11.
    J. A. Jenkins, Univalent Functions and Conformal Mapping [Russian translation], Moscow (1962).Google Scholar
  12. 12.
    V. N. Dubinin, “Symmetrization of condensers and inequalities for multivalent functions in a disk,” Mat. Zametki, 94, No. 6, 846–856 (2013).CrossRefMathSciNetGoogle Scholar
  13. 13.
    V. N. Dubinin, “On the Jenkins circles covering theorem for functions holomorphic in a disk,” Zap. Nauchn. Semin. POMI, 418, 60–73 (2013).Google Scholar
  14. 14.
    V. N. Dubinin, “A new version of the circular symmetrization with applications to p-valent functions,” Mat. Sb., 203, No. 7, 79–94 (2012).CrossRefMathSciNetGoogle Scholar
  15. 15.
    V. N. Dubinin, “Circular symmetrization of condensers on the Riemann surfaces,” Mat. Sb., 206, No. 2, 69–96 (2015).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Far Eastern Federal UniversityVladivostokRussia

Personalised recommendations