Advertisement

Mathematical Notes

, Volume 104, Issue 5–6, pp 683–688 | Cite as

Lemniscate Zone and Distortion Theorems for Multivalent Functions. II

  • V. N. DubininEmail author
Article

Abstract

For meromorphic circumferentially mean p-valent functions, an analog of the classical distortion theorem is proved. It is shown that the existence of connected lemniscates of the function and a constraint on a cover of two given points lead to an inequality involving the Green energy of a discrete signedmeasure concentrated at the zeros of the given function and the absolute values of its derivatives at these zeros. This inequality is an equality for the superposition of a certain univalent function and an appropriate Zolotarev fraction.

Keywords

meromorphic function p-valent function lemniscate Zolotarev fraction symmetrization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. N. Dubinin, “Lemniscate zone and distortion theorems for multivalent functions,” Zap. Nauchn. Sem. POMI 458, 17–30 (2017) [J.Math. Sci. 234 (5), 598–607 (2018)].Google Scholar
  2. 2.
    C. Carathéodory, “Sur quelques applications du théorè me de Landau–Picard,” Compt. Rend. Acad. Sci. 144, 1203–1206 (1907).zbMATHGoogle Scholar
  3. 3.
    G. M. Goluzin, The Geometric Theory of Functions of a Complex Variable (Nauka, Moscow, 1966) [in Russian].zbMATHGoogle Scholar
  4. 4.
    M. Biernacki, “Sur les fonctions en moyenne multivalentes,” Bull. Sci. Math. (2) 70, 51–76 (1946).MathSciNetzbMATHGoogle Scholar
  5. 5.
    W. K. Hayman, Multivalent Functions, in Cambridge Tracts in Math. (Cambridge Univ. Press, Cambridge, 1994), Vol.100.CrossRefzbMATHGoogle Scholar
  6. 6.
    J. A. Jenkins, Univalent Functions and Conformal Mapping (Springer-Verlag, Berlin–Heidelberg, 1958; Inostrannaya Literatura, Moscow, 1962).zbMATHGoogle Scholar
  7. 7.
    V. N. Dubinin, “Circular symmetrization of condensers on Riemann surfaces,” Mat. Sb. 206 (1), 69–96 (2015) [Sb.Math. 206 (1), 61–86 (2015)].MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    V. N. Dubinin, “A new version of circular symmetrization with applications to p-valent functions,” Mat. Sb. 203 (7), 79–94 (2012) [Sb.Math. 203 (7), 996–1011 (2012)].MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    V. N. Dubinin, “Symmetrization of condensers and inequalities for functions multivalent in a disk,” Mat. Zametki 94 (6), 846–856 (2013) [Math. Notes 94 (6), 876–884 (2013)].MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    V. N. Dubinin, “Distortion theorems for circumferentially mean p-valent functions,” Zap. Nauchn. Sem. POMI 440, 43–56 (2015) [J.Math. Sci. 217 (1), 28–36 (2016)].Google Scholar
  11. 11.
    N. S. Landkof, Foundations of Modern Potential Theory (Nauka, Moscow, 1966; Springer-Verlag, Berlin–Heidelberg, 1972). [in Russian].CrossRefzbMATHGoogle Scholar
  12. 12.
    V. N. Dubinin, “The logarithmic energy of zeros and poles of a rational function,” Sibirsk. Mat. Zh. 57 (6), 1255–1261 (2016) [SiberianMath. J. 57 (6), 981–986 (2016)].MathSciNetGoogle Scholar
  13. 13.
    V. N. Dubinin, “An extremal problem for the derivative of a rational function,” Mat. Zametki 100 (5), 732–738 (2016) [Math. Notes 100 (5), 714–719 (2016)].MathSciNetCrossRefGoogle Scholar
  14. 14.
    N. I. Akhiezer, Elements of the Theory of Elliptic Functions (Nauka, Moscow, 1970; Amer. Math. Soc., Providence, RI, 1990).CrossRefzbMATHGoogle Scholar
  15. 15.
    V. N. Dubinin, “Inequalities for themoduli of circumferentially mean p-valent functions,” Zap. Nauchn. Sem. POMI 429, 44–54 (2014) [J. Math. Sci. 207 (6), 832–838 (2015)].Google Scholar
  16. 16.
    V. N. Dubinin, Condenser Capacities and Symmetrization in Geometric Function Theory (Birkhäuser, Basel, 2014).CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Far-Eastern Federal UniversityVladivostokRussia
  2. 2.Institute for Applied Mathematics, Far-Eastern BranchRussian Academy of SciencesVladivostokRussia

Personalised recommendations