Skip to main content
SpringerLink
Log in

Asymptotic estimates for the coefficients of bounded univalent functions

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

In the class of univalent bounded normalized holomorphic functions in the unit disk, we give an asymptotic estimate for the coefficients when the uniform norm of the modulus of the function tends to infinity.

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

Access this article

Log in via an institution

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. De Branges L., A Proof of the Bieberbach Conjecture [Preprint E-5-84], LOMI (1984).

  2. De Branges L., “A proof of the Bieberbach conjecture,” Acta Math., 154, No. 1–2, 137–152 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  3. Prokhorov D., “Bounded univalent functions,” in: Handbook of Complex Analysis. Vol. 1: Geometric Function Theory, Elsevier Sci., Amsterdam, 2002, pp. 207–228.

    Google Scholar 

  4. Pick G., “Über die konforme Abbildung eines Kreises auf ein schlichtes und zugleich beschranktes Gebiet,” S.-B. Kaiserl. Akad. Wiss. Wien. Math.-Naturwiss. Kl. Abt. II a, Bd 126, 247–263 (1917).

    MATH  Google Scholar 

  5. Schaeffer A. C. and Spencer D. C., “The coefficients of schlicht functions. II,” Duke Math. J., 12, No. 1, 107–125 (1945).

    Article  MATH  MathSciNet  Google Scholar 

  6. Bazilevich I. E., “Domains of initial coefficients of bounded univalent functions with p-multiple symmetry,” Mat. Sb., 43, No. 4, 409–428 (1957).

    MathSciNet  Google Scholar 

  7. Schiffer M. and Tammi O., “On the fourth coefficient of bounded univalent functions,” Trans. Amer. Math. Soc., 119, No. 1, 67–78 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  8. Prokhorov D. and Vasil’ev A., “Optimal control in Bombieri’s and Tammi’s conjectures,” Georgian Math. J., 12, No. 4, 743–761 (2005).

    MATH  MathSciNet  Google Scholar 

  9. Prokhorov D., “Even coefficient estimates for bounded univalent functions,” Ann. Polon. Math., 58, No. 3, 267–273 (1993).

    MATH  MathSciNet  Google Scholar 

  10. Siewierski L., “Sharp estimation of the coefficients of bounded univalent functions near the identity,” Bull. Acad. Polon. Sci., 16, No. 7, 575–576 (1968).

    MATH  MathSciNet  Google Scholar 

  11. Siewierski L., “Sharp estimation of the coefficients of bounded univalent functions close to identity,” Dissertationes Math. (Rozprawy Mat.), 86, 1–153 (1971).

    MathSciNet  Google Scholar 

  12. Schiffer M. and Tammi O., “On bounded univalent functions which are close to identity,” Ann. Acad. Sci. Fenn. Ser. AI Math., 435, 3–26 (1968).

    MathSciNet  Google Scholar 

  13. Löwner K., “Untersuchungen uber schlichte konforme Abbildungen des Einheitskreises, ” Math. Ann., Bd 89, 103–121 (1923).

    Article  MATH  MathSciNet  Google Scholar 

  14. Prokhorov D. V., “Sets of values of systems of functionals in classes of univalent functions, ” Math. USSR-Sb., 71, No. 2, 499–516 (1992).

    Article  MathSciNet  Google Scholar 

  15. Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., and Mishchenko E. F., Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  16. Prokhorov D. V., “Geometric methods in a coefficient problem of analytic functions,” Izv. Saratov Univ. Novaya Ser., 1, No. 2, 43–55 (2001).

    MathSciNet  Google Scholar 

  17. Aleksandrov I. A., Parametric Extensions in the Theory of Univalent Functions [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  18. Bshouty D., “A coefficient problem of Bombieri concerning univalent functions,” Proc. Amer. Math. Soc., 91, No. 3, 383–388 (1984).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Saratov State University, Saratov, Russia

    D. V. Prokhorov & A. A. Nikulin

Authors
  1. D. V. Prokhorov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Nikulin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 5, pp. 1128–1138, September–October, 2006.

Original Russian Text Copyright © 2006 Prokhorov D. V. and Nikulin A. A.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Prokhorov, D.V., Nikulin, A.A. Asymptotic estimates for the coefficients of bounded univalent functions. Sib Math J 47, 924–933 (2006). https://doi.org/10.1007/s11202-006-0103-9

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11202-006-0103-9

Keywords

Search

Navigation