Third Hankel determinant for the inverse of a function whose derivative has a positive real part

Author
D. Vamshee Krishna, B. Venkateswarlu, T. RamReddy
Department of Mathematics, GIT, GITAM University, Visakhapatnam, India; Department of Mathematics, Kakatiya University, Warangal, India
Abstract
Let $RT$ be the class of functions $f$ univalent in the unit disk $E=\{z\colon |z|<1\}$ such that $\mathop{\rm Re}f'(z)>0,\ (z\in E)$ and $H_{3}(1)$ the third Hankel determinant for inverse function to $f\in RT.$ In the paper obtained the upper bound for $H_{3}(1)$ in the terms of Toeplitz determinants.
Keywords
analytic function; univalent function; function whose derivative has a positive real part; Hankel determinant; Toeplitz determinants
Reference
1. A. Abubaker, M. Darus, Hankel Determinant for a class of analytic functions involving a generalized linear differential operator, Int. J. Pure Appl. Math., 69 (2011), 4, 429435.

2. R.M. Ali, Coefficients of the inverse of strongly starlike functions, Bull. Malays. Math. Sci. Soc. (second series), 26 (2003), 1, 6371.

3. K.O. Babalola, On H3(1) Hankel determinant for some classes of univalent functions, Inequality Theory and Applications., 6 (2010), 17.

4. K.O. Babalola, T.O. Opoola, On the coefficients of certain analytic and univalent functions, Advances in Inequalities for series, Nova Science Publishers, (2008), 517.

5. D. Bansal, Upper bound of second Hankel determinant for a new class of analytic functions, Appl. Math. Lett., 26 (2013), 103107.

6. P.L. Duren, Univalent functions, V.259 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, USA, 1983.

7. R. Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly, 107 (2000), 6, 557560.

8. A.W. Goodman, Univalent functions, V.I, II, Mariner publishing Comp. Inc., Tampa, Florida, 1983.

9. U. Grenander, G. Szeg.o, Toeplitz forms and their applications, Second edition, Chelsea Publishing Co., New York, 1984.

10. W.K. Hayman, On the Second Hankel determinant of mean univalent functions, Proc. Lond. Math. Soc., 3 (1968), 7794.

11. A. Janteng, S.A. Halim, M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal. (Ruse), 1 (2007), 13, 619625.

12. A. Janteng, S.A. Halim, M. Darus, Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math., 7 (2006), 2, 15.

13. J.W. Layman, The Hankel transform and some of its properties, J. Integer Seq., 4 (2001), 1, 111.

14. R.J. Libera, E.J. Z lotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87 (1983), 251257.

15. R.J. Libera, E.J. Z lotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc., 85 (1982), 225230.

16. T.H. MacGregor, Functions whose derivative have a positive real part, Trans. Amer. Math. Soc., 104 (1962), 3, 532537.

17. J.W. Noonan, D.K. Thomas, On the second Hankel determinant of areally mean p-valent functions, Trans. Amer. Math. Soc., 223 (1976), 2, 337346.

18. K.I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roumaine Math. Pures Appl., 28 (1983), 8, 731739.

19. Ch. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, Gottingen, 1975.

20. Ch. Pommerenke, On the Hankel determinants of univalent functions, Mathematika, 14 (1967), 108112.

21. Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. Lond. Math. Soc., 41 (1966), 111122.

22. T. RamReddy, D. Vamshee Krishna, Hankel determinant for starlike and convex functions with respect to symmetric points, J. Indian Math. Soc., 79 (2012)(N. S.), 14, 161171.

23. B. Simon, Orthogonal polynomials on the unit circle, Part 1. Classical theory, American Mathematical Society Colloquium Publications, 54, Part 1, American Mathematical Society, Providence, RI, 2005.

24. D. Vamshee Krishna, T. RamReddy, An upper bound to the non-linear functional for certain subclasses of analytic functions associated with Hankel determinant, Asian-European J. Math., 7 (2014), 2, 114.

25. D. Vamshee Krishna, T. RamReddy, An upper bound to the second Hankel determinant for certain subclass of analytic functions, Proc. Jangeon Math. Soc., 16 (2013), 4, 559568.

26. D. Vamshee Krishna, T. RamReddy, Coefficient inequality for certain subclasses of analytic functions, New Zealand J. Math., 42 (2012), 217228.

27. S. Verma, S. Gupta, S. Singh, Bounds of Hankel determinant for a class of univalent functions, Int. J. Math. Sci., 2012, Article ID 147842, 6 p.

Pages
54-60
Volume
42
Issue
1
Year
2014
Journal
Matematychni Studii
Full text of paper
pdf
Table of content of issue