Advertisement

Results in Mathematics

, 73:89 | Cite as

On Hankel Determinant \({{\varvec{H}}}_\mathbf{2}{} \mathbf{(3)}\) for Univalent Functions

Open Access
Article
  • 12 Downloads

Abstract

In this paper we consider the Hankel determinant \(H_2(3) = a_3a_5 - a_4{}^2\) defined for the coefficients of a function f which belongs to the class \(\mathcal {S}\) of univalent functions or to its subclasses: \(S^*\) of starlike functions, \(\mathcal {K}\) of convex functions and \(\mathcal {R}\) of functions whose derivative has a positive real part. Bounds of \(|H_2(3)|\) for these classes are found; the bound for \(\mathcal {R}\) is sharp. Moreover, the sharp results for starlike functions and convex functions for which \(a_2=0\) are obtained. It is also proved that \(\max \{|H_2(3)|: f\in \mathcal {S}\}\) is greater than 1.

Keywords

Univalent functions starlike functions convex functions Hankel determinant 

Mathematics Subject Classification

30C50 

References

  1. 1.
    Bansal, D., Maharana, S., Prajapat, J.K.: Third order Hankel determinant for certain univalent functions. J. Korean Math. Soc. 52(6), 1139–1148 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brown, J.E.: Successive coefficients of functions with positive real part. Int. J. Math. Anal. 4(50), 2491–2499 (2010)MathSciNetMATHGoogle Scholar
  3. 3.
    Cho, N.E., Kowalczyk, B., Kwon, O.S., Lecko, A., Sim, Y.J.: Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha. J. Math. Inequal. 11(2), 429–439 (2017)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Deniz, E., Caglar, M., Orhan, H.: Second Hankel determinant for bi-starlike and bi-convex functions of order beta. Appl. Math. Comput. 271, 301–307 (2015)MathSciNetGoogle Scholar
  5. 5.
    Grenander, U., Szegö, G.: Toeplitz Forms and Their Application. University of Calofornia Press, Berkely (1958)MATHGoogle Scholar
  6. 6.
    Hayami, T., Owa, S.: Generalized Hankel determinant for certain classes. Int. J. Math. Anal. 4(49–52), 2573–2585 (2010)MathSciNetMATHGoogle Scholar
  7. 7.
    Jakubowski, Z.J.: On some extremal problems of the theory of univalent functions. In: Kühnau, R. et al. (eds.) Geometric Function Theory and Applications of Complex Analysis and Its Applications to Partial Differential Equations 2, pp. 49–55. Pitman Research Notes in Mathematics Series 257 (1993)Google Scholar
  8. 8.
    Janteng, A., Halim, S.A., Darus, M.: Coefficient inequality for a function whose derivative has a positive real part. J. Inequal. Pure Appl. Math. 7(2), 1–5 (2006)MathSciNetMATHGoogle Scholar
  9. 9.
    Janteng, A., Halim, S.A., Darus, M.: Hankel determinant for starlike and convex functions. Int. J. Math. Anal. Ruse 1(13–16), 619–625 (2007)MathSciNetMATHGoogle Scholar
  10. 10.
    Lee, S.K., Ravichandran, V., Supramaniam, S.: Bounds for the second Hankel determinant of certain univalent functions. J. Inequal. Appl. 2013, 281 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Li, J.-L., Srivastava, H.M.: Some questions and conjectures in the theory of univalent functions. Rocky Mt. J. Math. 28(3), 1035–1041 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Libera, R.J., Złotkiewicz, E.J.: Early coefficients of the inverse of a regular convex function. Proc. Am. Math. Soc. 85, 225–230 (1982)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Livingston, A.E.: The coefficients of multivalent close-to-convex functions. Proc. Am. Math. Soc. 21, 545–552 (1969)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Marjono, Thomas, D.K.: The second Hankel determinant of functions convex in one direction. Int. J. Math. Anal. 10(9), 423–428 (2016)CrossRefMATHGoogle Scholar
  15. 15.
    Mishra, A.K., Prajapat, J.K., Maharana, S.: Bounds on Hankel determinant for starlike and convex functions with respect to symmetric points. Cogent Math. (2016).  https://doi.org/10.1080/23311835.2016.1160557 MathSciNetGoogle Scholar
  16. 16.
    Noonan, J.W., Thomas, D.K.: On the Hankel determinants of areally mean p-valent functions. Proc. Lond. Math. Soc. 25, 503–524 (1972)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Orhan, H., Magesh, N., Yamini, J.: Bounds for the second Hankel determinant of certain bi-univalent functions. Turkish J. Math. 40(3), 679–687 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Parvatham, R., Ponnusamy, S. (eds.): New Trends in Geometric Function Theory and Application. World Scientific Publishing Company, Singapore (1981)Google Scholar
  19. 19.
    Pommerenke, C.: On the coefficients and Hankel determinants of univalent functions. J. Lond. Math. Soc. 41, 111–122 (1966)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Pommerenke, C.: On the Hankel determinants of univalent functions. Mathematika 14, 108–112 (1967)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Răducanu, D., Zaprawa, P.: Second Hankel determinant for close-to-convex functions. C. R. Math. Acad. Sci. Paris 355(10), 1063–1071 (2017)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Raza, M., Malik, S.N.: Upper bound of third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli. J. Inequal. Appl. 2013, 412 (2013)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Schaeffer, A.C., Spencer, D.C.: The coefficients of schlicht functions. Duke Math. J. 10, 611–635 (1943)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Vamshee Krishna, D., Venkateswarlua, B., RamReddy, T.: Third Hankel determinant for bounded turning functions of order alpha. J. Niger. Math. Soc. 34, 121–127 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Zaprawa, P.: Third Hankel determinants for subclasses of univalent functions. Mediterr. J. Math. 14(1), 19 (2017)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mechanical EngineeringLublin University of TechnologyLublinPoland

Personalised recommendations