Skip to main content
Log in

Hermitian-Toeplitz determinants and some coefficient functionals for the starlike functions

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we have determined the sharp lower and upper bounds on the fourth-order Hermitian-Toeplitz determinant for starlike functions with real coefficients. We also obtained the sharp bounds on the Hermitian-Toeplitz determinants of inverse and logarithmic coefficients for starlike functions with complex coefficients. Sharp bounds on the modulus of differences and difference of moduli of logarithmic and inverse coefficients are obtained. In our investigation, it has been found that the bound on the third-order Hermitian-Toeplitz determinant for starlike functions and its inverse coefficients is invariant.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /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.

Similar content being viewed by others

References

  1. M. F. Ali, D. K. Thomas, A. Vasudevarao: Toeplitz determinants whose elements are the coefficients of analytic and univalent functions. Bull. Aust. Math. Soc. 97 (2018), 253–264.

    Article  MathSciNet  MATH  Google Scholar 

  2. K. O. Babalola: On H3,1 Hankel determinants for some classes of univalent functions. Inequality Theory & Applications 6. Nova Science Publishers, New York, 2010, pp. 1–7.

    Google Scholar 

  3. N. E. Cho, V. Kumar, O. S. Kwon, Y. J. Sim: Coefficient bounds for certain subclasses of starlike functions. J. Inequal. Appl. 2019 (2019), Article ID 276, 13 pages.

    Article  MathSciNet  MATH  Google Scholar 

  4. K. Cudna, O. S. Kwon, A. Lecko, Y. J. Sim, B. Śmiarowska: The second and third-order Hermitian Toeplitz determinants for starlike and convex functions of order α. Bol. Soc. Mat. Mex., III. Ser. 26 (2020), 361–375.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. L. Duren: Univalent Functions. Grundlehren der Mathematischen Wissenschaften 259. Springer, New York, 1983.

    Google Scholar 

  6. W. K. Hayman: On the second Hankel determinant of mean univalent functions. Proc. Lond. Math. Soc., III. Ser. 18 (1968), 77–94.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Janteng, S. A. Halim, M. Darus: Hankel determinant for starlike and convex functions. Int. J. Math. Anal., Ruse 1 (2007), 619–625.

    MathSciNet  MATH  Google Scholar 

  8. P. Jastrzȩbski, B. Kowalczyk, O. S. Kwon, Y. J. Sim: Hermitian Toeplitz determinants of the second and third-order for classes of close-to-star functions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 114 (2020), Article ID 166, 14 pages.

    MathSciNet  MATH  Google Scholar 

  9. B. Kowalczyk, A. Lecko: Second Hankel determinant of logarithmic coefficients of convex and starlike functions. Bull. Aust. Math. Soc. 105 (2022), 458–467.

    Article  MathSciNet  MATH  Google Scholar 

  10. B. Kowalczyk, A. Lecko: Second Hankel determinant of logarithmic coefficients of convex and starlike functions of order alpha. Bull. Malays. Math. Sci. Soc. (2) 45 (2022), 727–740.

    Article  MathSciNet  MATH  Google Scholar 

  11. B. Kowalczyk, A. Lecko, Y. J. Sim: The sharp bound for the Hankel determinant of the third kind for convex functions. Bull. Aust. Math. Soc. 97 (2018), 435–445.

    Article  MathSciNet  MATH  Google Scholar 

  12. V. Kumar: Hermitian-Toeplitz determinants for certain classes of close-to-convex functions. Bull. Iran. Math. Soc. 48 (2022), 1093–1109.

    Article  MathSciNet  MATH  Google Scholar 

  13. V. Kumar, N. E. Cho: Hermitian-Toeplitz determinants for functions with bounded turning. Turk. J. Math. 45 (2021), 2678–2687.

    Article  MathSciNet  MATH  Google Scholar 

  14. V. Kumar, S. Kumar: Bounds on Hermitian-Toeplitz and Hankel determinants for strongly starlike functions. Bol. Soc. Mat. Mex., III. Ser. 27 (2021), Article ID 55, 16 pages.

    MathSciNet  MATH  Google Scholar 

  15. V. Kumar, S. Nagpal, N. E. Cho: Coefficient functionals for non-Bazilevič functions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 116 (2022), Article ID 44, 14 pages.

    MATH  Google Scholar 

  16. V. Kumar, R. Srivastava, N. E. Cho: Sharp estimation of Hermitian-Toeplitz determinants for Janowski type starlike and convex functions. Miskolc Math. Notes 21 (2020), 939–952.

    Article  MathSciNet  MATH  Google Scholar 

  17. O. S. Kwon, A. Lecko, Y. J. Sim: The bound of the Hankel determinant of the third kind for starlike functions. Bull. Malays. Math. Sci. Soc. (2) 42 (2019), 767–780.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Lecko, Y. J. Sim, B. Śmiarowska: The fourth-order Hermitian Toeplitz determinant for convex functions. Anal. Math. Phys. 10 (2020), Article ID 39, 11 pages.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. K. Lee, V. Ravichandran, S. Supramaniam: Bounds for the second Hankel determinant of certain univalent functions. J. Inequal. Appl. 2013 (2013), Article ID 281, 17 pages.

    Article  MathSciNet  MATH  Google Scholar 

  20. R. J. Libera, E. J. Złotkiewicz: Early coefficients of the inverse of a regular convex function. Proc. Am. Math. Soc. 85 (1982), 225–230.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. J. Libera, E. J. Złotkiewicz: Coefficient bounds for the inverse of a function with derivative in P. Proc. Am. Math. Soc. 87 (1983), 251–257.

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  23. Y. J. Sim, D. K. Thomas: On the difference of inverse coefficients of univalent functions. Symmetry 12 (2020), Article ID 2040, 14 pages.

    Article  Google Scholar 

  24. D. K. Thomas: On the coefficients of strongly starlike functions. Indian J. Math. 58 (2016), 135–146.

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to express their gratitude to the referees for many valuable suggestions regarding the previous version of this paper, which indeed improved the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Virendra Kumar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, D., Kumar, V. & Das, L. Hermitian-Toeplitz determinants and some coefficient functionals for the starlike functions. Appl Math 68, 289–304 (2023). https://doi.org/10.21136/AM.2022.0092-22

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/AM.2022.0092-22

Keywords

MSC 2020

Navigation