Skip to main content
Log in

On the Second Hankel Determinant of Logarithmic Coefficients for Certain Univalent Functions

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we investigate the sharp bounds of the second Hankel determinant of Logarithmic coefficients for the starlike and convex functions with respect to symmetric points in the open unit disk.

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.

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Ali, M.F., Allu, V.: On logarithmic coefficients of some close-to-convex functions. Proc. Amer. Math. Soc. 146, 1131–1142 (2017)

    Article  MathSciNet  Google Scholar 

  2. Ali, M.F., Allu, V.: Logarithmic coefficients of some close-to-convex functions. Bull. Aust. Math. Soc. 95, 228–237 (2017)

    Article  MathSciNet  Google Scholar 

  3. Allu, V., Lecko, A., Thomas, D. K.: Hankel, Toeplitz and Hermitian-Toeplitz Determinants for Ozaki Close-to-convex Functions, Mediterr. J. Math. (to appear)

  4. Allu, V., Arora, V.: Second Hankel determinant of logarithmic coefficients of certain analytic functions, arXiv: 2110.05161

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Cho, N.E., Kowalczyk, B., Kwon, O., Lecko, A., Sim, Y.: On the third logarithmic coefficient in some subclasses of close-to-convex functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. 114, (2020). https://doi.org/10.1007/s13398-020-00786-7

  7. Das, R.N., Singh, P.: On subclasses of schlicht mapping. Indian J. Pure Appl. Math. 8, 864–872 (1977)

    MathSciNet  MATH  Google Scholar 

  8. Duren, P.L.: Univalent functions (Grundlehren der mathematischen Wissenschaften 259, New York, Berlin, Heidelberg, Tokyo), Springer-Verlag, (1983)

  9. Efraimidis, I.: A generalization of Livingston’s coefficient inequalities for functions with positive real part. J. Math. Anal. Appl. 435, 369–379 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Elhosh, M.M.: On the logarithmic coefficients of close-to-convex functions. J. Aust. Math. Soc. A 60, 1–6 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Girela, D.: Logarithmic coefficients of univalent functions. Ann. Acad. Sci. Fenn. Math. 35(2), 337–350 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Kowalczyk, B., Lecko, A.: Second hankel determinant of logarithmic coefficients of convex and starlike functions, Bull. Aust. Math. Soc. https://doi.org/10.1017/S0004972721000836

  13. Milin, I. M.: Univalent functions and orthonormal systems, Translations of Mathematical Monographs, Volume 49 (1977)

  14. Nezhmetdinov, I.R., Ponnusamy, S.: On the class of univalent functions starlike with respect to N-symmetric points. Hokkaido Math. J. 31(1), 61–77 (2002)

    MathSciNet  MATH  Google Scholar 

  15. Pommerenke, Ch.: On the coefficients and Hankel determinants of univalent functions. J. Lond. Math. Soc. 41, 111–122 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pommerenke, Ch.: On the Hankel determinants of Univalent functions. Mathematika 14, 108–112 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ponnusamy, S., Sharma, N.L., Wirths, K.J.: Logarithmic coefficients problems in families related to starlike and convex functions. J. Aust. Math. Soc. 109, 230–249 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ponnusamy, S., Sugawa, T.: Sharp inequalities for logarithmic coefficients and their applications, Bulletin des Sciences Mathématiques 166, 23 pages, Article 102931 (2021)

  19. Pranav Kumar, U., Vasudevarao, A.: Logarithmic coefficients for certain subclasses of close-to-convex functions. Monatsh. Math. 187, 543–563 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sakaguchi, K.: On a certain univalent mapping. J. Math. Soc. Japan 11, 72–75 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sim, Y.J., Lecko, A., Thomas, D.K.: The second Hankel determinant for strongly convex and Ozaki close-to-convex functions. Ann. Mat. Pura. Appl. 200, 2515–2533 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  22. Thomas, D.K.: On the logarithmic coefficients of close-to-convex functions. Proc. Amer. Math. Soc. 144, 1681–1687 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zaprawa, P.: Initial logarithmic coefficients for functions starlike with respect to symmetric points. Bol. Soc. Mat. Mex. 27, (2021). https://doi.org/10.1007/s40590-021-00370-y

Download references

Acknowledgements

The first author thanks SERB-CRG, the second author thanks IIT Bhubaneswar for providing Institute Post Doctoral Fellowship, and the third author’s research work is supported by CSIR-UGC.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vasudevarao Allu.

Ethics declarations

Conflict of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

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., Arora, V. & Shaji, A. On the Second Hankel Determinant of Logarithmic Coefficients for Certain Univalent Functions. Mediterr. J. Math. 20, 81 (2023). https://doi.org/10.1007/s00009-023-02272-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-023-02272-x

Keywords

Mathematics Subject Classification

Navigation