Skip to main content
Log in

Inequalities for the Moduli of Circumferentially mean p-Valent Functions

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Montel, Leçons sur les Fonctions Univalentes ou Multivalentes, Gauthier–Villars, Paris (1933).

    MATH  Google Scholar 

  2. J. Krzyz, “On univalent functions with two preassigned values,” Ann. Univ. Mariae Curie-Sklodowska, Sec. A, 15, No. 5, 57–77 (1961).

    MATH  MathSciNet  Google Scholar 

  3. A. Vasil’ev, “Moduli of families of curves for conformal and quasiconformal mappings,” Lect. Notes Math., 1788, Springer (2002).

  4. V. Singh, “Some extremal problems for a new class of univalent functions,” J. Math. Mech., 7, 811–821 (1958).

    MATH  MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  6. J. Krzyz and E. Zlotkiewicz, “Koebe sets for univalent functions with two preassigned values,” Ann. Acad. Sci. Fenn., Ser. A1. Math., 487 (1971).

  7. R. J. Libera and E. J. Zlotkiewicz, “Bounded Montel univalent functions,” Colloq. Math., 56, 169–177 (1988).

    MATH  MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  10. W. K. Hayman, Multivalent functions, 2nd ed., Cambridge Univ. Press, Cambridge (1994).

    Book  MATH  Google Scholar 

  11. J. A. Jenkins, Univalent Functions and Conformal Mapping [Russian translation], Moscow (1962).

  12. V. N. Dubinin, “Symmetrization of condensers and inequalities for multivalent functions in a disk,” Mat. Zametki, 94, No. 6, 846–856 (2013).

    Article  MathSciNet  Google Scholar 

  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. V. N. Dubinin, “A new version of the circular symmetrization with applications to p-valent functions,” Mat. Sb., 203, No. 7, 79–94 (2012).

    Article  MathSciNet  Google Scholar 

  15. V. N. Dubinin, “Circular symmetrization of condensers on the Riemann surfaces,” Mat. Sb., 206, No. 2, 69–96 (2015).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. N. Dubinin.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 429, 214, pp. 44–54.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dubinin, V.N. Inequalities for the Moduli of Circumferentially mean p-Valent Functions. J Math Sci 207, 832–838 (2015). https://doi.org/10.1007/s10958-015-2407-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-015-2407-4

Keywords

Navigation