# Article

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Keywords:
analytic function; upper bound; second Hankel functional; positive real function; Toeplitz determinant
Summary:
The objective of this paper is to obtain sharp upper bound for the function $f$ for the second Hankel determinant $|a_{2}a_{4}-a_{3}^{2}|$, when it belongs to the class of functions whose derivative has a positive real part of order $\alpha$ $(0\leq \alpha <1)$, denoted by $RT(\alpha )$. Further, an upper bound for the inverse function of $f$ for the nonlinear functional (also called the second Hankel functional), denoted by $|t_{2}t_{4}-t_{3}^{2}|$, was determined when it belongs to the same class of functions, using Toeplitz determinants.
References:
[1] Abubaker, A., Darus, M.: Hankel determinant for a class of analytic functions involving a generalized linear differential operator. Int. J. Pure Appl. Math. 69 429-435 (2011). MR 2847841 | Zbl 1220.30011
[2] Ali, R. M.: Coefficients of the inverse of strongly starlike functions. Bull. Malays. Math. Sci. Soc. (2) 26 63-71 (2003). MR 2055766 | Zbl 1185.30010
[3] Ehrenborg, R.: The Hankel determinant of exponential polynomials. Am. Math. Mon. 107 557-560 (2000). DOI 10.2307/2589352 | MR 1767065 | Zbl 0985.15006
[4] Grenander, U., Szegő, G.: Toeplitz Forms and Their Applications. Chelsea Publishing Co., New York (1984). MR 0890515 | Zbl 0611.47018
[5] Janteng, A., Halim, S. A., Darus, M.: Hankel determinant for starlike and convex functions. Int. J. Math. Anal., Ruse 1 619-625 (2007). MR 2370200 | Zbl 1137.30308
[6] Janteng, A., Halim, S. A., Darus, M.: Coefficient inequality for a function whose derivative has a positive real part. J. Inequal. Pure Appl. Math. (electronic only) 7 Article 50, 5 pages (2006). MR 2221331 | Zbl 1134.30310
[7] Layman, J. W.: The Hankel transform and some of its properties. J. Integer Seq. (electronic only) 4 Article 01.1.5, 11 pages (2001). MR 1848942 | Zbl 0978.15022
[8] MacGregor, T. H.: Functions whose derivative has a positive real part. Trans. Am. Math. Soc. 104 532-537 (1962). DOI 10.1090/S0002-9947-1962-0140674-7 | MR 0140674 | Zbl 0106.04805
[9] Mishra, A. K., Gochhayat, P.: Second Hankel determinant for a class of analytic functions defined by fractional derivative. Int. J. Math. Math. Sci. 2008 Article ID 153280, 10 pages (2008). MR 2392999 | Zbl 1158.30308
[10] Murugusundaramoorthy, G., Magesh, N.: Coefficient inequalities for certain classes of analytic functions associated with Hankel determinant. Bull. Math. Anal. Appl. 1 85-89 (2009). MR 2578118 | Zbl 1312.30024
[11] Noonan, J. W., Thomas, D. K.: On the second Hankel determinant of areally mean $p$-valent functions. Trans. Am. Math. Soc. 223 337-346 (1976). MR 0422607 | Zbl 0346.30012
[12] Noor, K. I.: Hankel determinant problem for the class of functions with bounded boundary rotation. Rev. Roum. Math. Pures Appl. 28 731-739 (1983). MR 0725316 | Zbl 0524.30008
[13] Pommerenke, C.: Univalent Functions. With a Chapter on Quadratic Differentials by Gerd Jensen. Studia Mathematica/Mathematische Lehrbücher. Band 25 Vandenhoeck & Ruprecht, Göttingen (1975). MR 0507768 | Zbl 0298.30014
[14] Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. American Mathematical Society Colloquium Publications 54 AMS, Providence (2005). DOI 10.1090/coll/054.2 | MR 2105088 | Zbl 1082.42020

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