Skip to main content
SpringerLink
Log in

A Problem of Bombieri on Univalent Functions

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

The famous Bieberbach Conjecture from 1916 on the coefficients of normalized univalent functions defined in the unit disk (Bieberbach, S.-B. Preuss Akad Wiss 138:940–955, 1916) that was finally proved by de Branges (Acta Math 154:137–152, 1985) some 70 years later, diverted the attention of many complex analysts to other subjects. Those who continued to explore de Branges method and push it as far as possible were not aware of where it may lead. Surprisingly enough, a paper that fell in our hands (Dong Acta Sci Nat Univ Norm Hunan 14:193–197, 1991) contained a way to tackle one of the problems of Bombieri (Research problems in function theory, The Athlone Press, University of London, London, 1967) on the behavior of the coefficients of univalent functions. We shall give an account of the history of the problem and a revised version of it.

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. Aharonov, D.: Special topics in univalent functions. University of Maryland, Lecture Notes (1971)

  2. Aharonov, D.: On the Bieberbach conjecture for functions with a small second coefficient. Israel J. Math. 15, 137–139 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bieberbach, L.: Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. S.-B. Preuss. Akad. Wiss. 138, 940–955 (1916)

    MATH  Google Scholar 

  4. Bombieri, E.: On the local maximum property of the Koebe function. Invent. Math. 4, 26–67 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  5. de Branges, L.: A proof of the Bieberbach conjecture. Acta Math. 154, 137–152 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bshouty, D.: The Bieberbach conjecture for restricted initial coefficients. Math. Z. 182, 149–158 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bshouty, D.: A coefficient problem of Bombieri concerning univalent functions. Proc. Amer. Math. Soc. 91, 383–388 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bshouty, D., Hengartner, W.: A variation of the Koebe mapping in a dense subset of S. Can. J. Math. 39, 54–73 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Charzyński, Z., Schiffer, M.: A geometric proof of the Bieberbach conjecture for the fourth coefficient. Scripta Math. 25, 173–181 (1960)

    MathSciNet  MATH  Google Scholar 

  10. Dong, X.H.: A remark on de Branges theorem, Acta Sci. Natur. Univ. Norm. Hunan. 14, 193–197 (1991) (205)

  11. Duren, P.L., Schiffer, M.: The theory of the second variation in extremum problems for univalent functions, J. Analyse Math. 10, 193–252 (1962/1963)

  12. Ehrig, G.: The Bieberbach conjecture for univalent functions with restricted second coefficients. J. London Math. Soc. 8(2), 355–360 (1974)

  13. FitzGerald, C.H.: Quadratic inequalities and coefficient estimates for schlicht functions. Arch. Ration. Mech. Anal. 46, 356–368 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fitzgerald, C.: The Bieberbach conjecture: retrospective. Notices Amer. Math. Soc. 32, 2–6 (1985)

    MathSciNet  MATH  Google Scholar 

  15. Garabedian, P.R., Schiffer, M.: A proof of the Bieberbach conjecture for the fourth coefficient. J. Ration. Mech. Anal. 4, 427–465 (1955)

    MathSciNet  MATH  Google Scholar 

  16. Garabedian, P.R., Schiffer, M.: The local maximum theorem for the coefficients of univalent functions. Arch. Ration. Mech. Anal. 26, 1–32 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  17. Garabedian, P.R., Ross, G.G., Schiffer, M.M.: On the Bieberbach conjecture for even n. J. Math. Mech. 14, 975–989 (1965)

    MathSciNet  MATH  Google Scholar 

  18. Goluzin, G.M.: Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, vol. 26. American Mathematical Society, Providence, R.I. (1969)

  19. Górski, J.: A certain minimum problem in the class S, Ann. Univ. Mariae Curie-Skodowska Sect. A 22–24, 73–77 (1968–1970)

  20. Greiner, R., Roth, O.: On support points of univalent functions and a disproof of a conjecture of Bombieri. Proc. Amer. Math. Soc. 129, 3657–3664 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Grunsky, H.: Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen. Math. Z. 45, 29–61 (1939)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hayman, W.K.: Research problems in function theory. The Athlone Press, University of London, London (1967)

    MATH  Google Scholar 

  23. Hummel, J.A.: The coefficients of starlike functions. Proc. Amer. Math. Soc. 22, 311–315 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, J.-L.: Notes on the Duren-Leung conjecture. J. Math. Anal. Appl. 332, 164–170 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Löwner, K.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises I. Math. Ann. 89, 103–121 (1923)

    Article  MathSciNet  MATH  Google Scholar 

  26. Milin, I.M.: The coefficients of schlicht functions, (Russian) Dokl. Akad. Nauk SSSR 176, 1015–1018 (1967). English transl. [Soviet Math. Dokl. 8 (1967), 1255–1258]

  27. Milin, I.M.: Univalent functions and orthonormal systems, (Russian) Izdat. Nauka, Moscow, (1971) English transl. [Math. Monos., vol. 50, Amer. Math. Soc., Providence, R.I., 1977.]

  28. Pommerenke, Ch.: Univalent Functions. Vandenhoeck and Ruprecht, Gttingen (1975)

    MATH  Google Scholar 

  29. Prokhorov, D., Roth, O.: On the local extremum property of the Koebe function. Math. Proc. Cambridge Philos. Soc. 136, 301–312 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Prokhorov, D., Vasilev, A.: Optimal control in Bombieri’s and Tammi’s conjectures. Georgian Math. J. 12, 743–761 (2005)

    MathSciNet  MATH  Google Scholar 

  31. Schiffer, M.: Sur un problème d’extrémum de la représentation conforme. Bull. Soc. Math. France 66, 48–55 (1938)

    MathSciNet  MATH  Google Scholar 

  32. Schiffer, M.: A method of variation within the family of simple functions. Proc. London Math. Soc. S2–44, 432–449 (1938)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the referees for careful reading of the manuscript which improved its content and presentation.

Author information

Authors and Affiliations

  1. Department of Mathematics, Technion IIT, Haifa, 32000, Israel

    D. Aharonov & D. Bshouty

Authors
  1. D. Aharonov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Bshouty
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Bshouty.

Additional information

Communicated by Stephan Ruscheweyh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aharonov, D., Bshouty, D. A Problem of Bombieri on Univalent Functions. Comput. Methods Funct. Theory 16, 677–688 (2016). https://doi.org/10.1007/s40315-016-0165-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40315-016-0165-z

Keywords

Mathematics Subject Classification

Search

Navigation