Skip to main content
Springer Nature Link
Log in

On the generalized Zalcman conjecture

  • Published:
Annali di Matematica Pura ed Applicata (1923 -) Aims and scope Submit manuscript

Abstract

Let \(\mathcal {S}\) denote the class of analytic and univalent (i.e., one-to-one) functions \( f(z)= z+\sum _{n=2}^{\infty }a_n z^n\) in the unit disk \(\mathbb {D}=\{z\in \mathbb {C}:|z|<1\}\). For \(f\in \mathcal {S}\), In 1999, Ma proposed the generalized Zalcman conjecture that

$$\begin{aligned}|a_{n}a_{m}-a_{n+m-1}|\le (n-1)(m-1),\,\,\, \text{ for } n\ge 2,\, m\ge 2,\end{aligned}$$

with equality only for the Koebe function \(k(z) = z/(1 - z)^2\) and its rotations. In the same paper, Ma (J Math Anal Appl 234:328–339, 1999) asked for what positive real values of \(\lambda \) does the following inequality hold?

$$\begin{aligned} |\lambda a_na_m-a_{n+m-1}|\le \lambda nm -n-m+1 \,\,\,\,\, (n\ge 2, \,m\ge 3). \end{aligned}$$
(0.1)

Clearly equality holds for the Koebe function \(k(z) = z/(1 - z)^2\) and its rotations. In this paper, we prove the inequality (0.1) for \(\lambda =3, n=2, m=3\). Further, we provide a geometric condition on extremal function maximizing (0.1) for \(\lambda =2,n=2, m=3\).

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Ahlfors, L.V.: Conformal Invariants. AMS chelsa Publishing, New York (1973)

    Google Scholar 

  2. Bombieri, E.: A geometric approach to some coefficient inequalities for univalent functions. Ann. Scoula Norm. Sup. Pisa 22, 377–397 (1968)

    MathSciNet  Google Scholar 

  3. Brown, J.E., Tsao, A.: On the Zalcman conjecture for starlike and typically real functions. Math. Z. 191, 467–474 (1986)

    Article  MathSciNet  Google Scholar 

  4. Charzynski, Z., Schiffer, M.: A new proof of the Bieberbach conjecture for the 4th coefficient. Arch. Ration. Mech. Anal. 5, 187–193 (1960)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Duren, P.L.: Univalent Functions (Grundlehren der mathematischen Wissenschaften 259, Berlin, Heidelberg, Tokyo). Springer, New York (1983)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  8. Jenkins, J.A.: On certain coefficients of univalent functions. In: Analytic Functions. Princeton Univ. Press, pp. 159–164 (1960)

  9. Jenkins, J.A.: Univalent Functions and Conformal Mappings. Springer, Berlin (1965)

    Google Scholar 

  10. Krushkal, S.L.: Univalent functions and holomorphic motions. J. Anal. Math. 66, 253–275 (1995)

    Article  MathSciNet  Google Scholar 

  11. Krushkal, S.L.: Proof of the Zalcman conjecture for initial coefficients. Georgian Math. J. 17, 663–681 (2010)

    Article  MathSciNet  Google Scholar 

  12. Krushkal, S.L.: A short geometric proof of the Zalcman conjecture and Bieberbach conjecture, preprint, arXiv:1408.1948

  13. Li, L., Ponnusamy, S.: On the generalized Zalcman functional \(\lambda a_n^2-a_{2n-1}\) in the close-to-convex family. Proc. Am. Math. Soc. 145, 833–846 (2017)

    Article  Google Scholar 

  14. Ma, W.: The Zalcman conjecture for close-to-convex functions. Proc. Am. Math. Soc. 104, 741–744 (1988)

    Article  MathSciNet  Google Scholar 

  15. Ma, W.: Generalized Zalcman conjecture for starlike and typically real functions. J. Math. Anal. Appl. 234, 328–339 (1999)

    Article  MathSciNet  Google Scholar 

  16. Ozawa, M.: On Certain coefficient inequalities for uniavlent functions. Kodai Math. Sem. Rep. 16, 183–188 (1964)

    Article  Google Scholar 

  17. Pfluger, A.: On a coefficient problem for Schlicht functions. In: Advances in Complex Function Theory (Proc. Sem., Univ. Maryland, College Park, Md., 1973), Lecture Notes in Math., Vol. 505. Springer, Berlin, pp. 79–91 (1976)

  18. Schaeffer, A.C., Spencer, D.C.: The coefficients of Schilcht functions-I. Duke Math. J. 10, 611–635 (1943)

    Article  MathSciNet  Google Scholar 

  19. Schaeffer, A.C., Spencer, D.C.: The coefficients of Schilcht functions-II. Duke Math. J. 12, 107–125 (1945)

    Article  MathSciNet  Google Scholar 

  20. Schaeffer, A.C., Spencer, D.C.: The coefficients of Schilcht functions-III. Proc. Natl. Acad. Sci. 32, 111–116 (1946)

    Article  Google Scholar 

  21. Schaeffer, A.C., Spencer, D.C.: The coefficients of Schilcht functions-IV. Proc. Natl. Acad. Sci. 35, 143–150 (1949)

    Article  Google Scholar 

  22. Schaeffer, A.C., Schiffer, M., Spencer, D.C.: The coefficients of Schilcht functions. Duke Math. J. 16, 493–527 (1949)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Schiffer, M.: On the coefficients of simple functions. Proc. Lond. Math. Soc. 44, 450–452 (1938)

    Article  MathSciNet  Google Scholar 

  25. Schiffer, M.: On the coefficient problem for univalent functions. Trans. Am. Math. Soc. 134(1938), 95–101 (1968)

    Article  MathSciNet  Google Scholar 

  26. Strebel, K.: Quadratic Differential. Springer, Berlin (1984)

    Book  Google Scholar 

Download references

Acknowledgements

Authors thanks Prof. Hiroshi Yanagihara and Prof. D. K. Thomas for fruitful discussions and giving constructive suggestions for improvements to this paper. The first author thanks SERB-CRG, and the second author thanks Prime Minister’s Research Fellowship (Id: 1200297) for their support.

Author information

Authors and Affiliations

  1. Discipline of Mathematics, School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Argul, Bhubaneswar, Odisha, 752050, India

    Vasudevarao Allu & Abhishek Pandey

Authors
  1. Vasudevarao Allu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abhishek Pandey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vasudevarao Allu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Allu, V., Pandey, A. On the generalized Zalcman conjecture. Annali di Matematica 203, 2665–2675 (2024). https://doi.org/10.1007/s10231-024-01461-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10231-024-01461-z

Keywords

Mathematics Subject Classification

Search

Navigation