The sharp bound of the third Hankel determinants for inverse of starlike functions with respect to symmetric points
Abstract
We study the sharp bound for the third Hankel determinant for the inverse function $f$, when it belongs to of the class of starlike functions with respect to symmetric points.
Let $\mathcal{S}^{\ast}_{s}$ be the class of starlike functions with respect to symmetric points. In the article proves the following statements (Theorem): If $f\in \mathcal{S}^{\ast}_{s}$ then
\begin{equation*}
\big|H_{3,1}(f^{-1})\big|\leq1,
\end{equation*}
and the result is sharp for $f(z)=z/(1-z^2).$
References
R.M. Ali, Coefficient of the inverse of strongly starlike functions, Bull. Malayasian Math. Sci. Soc. (second series), 26 (2003), 63–71.
K.O. Babalola, On H3(1) Hankel determinant for some classes of univalent functions, Inequality Theory and Applications, 6 (ed. Y. J. Cho)(Nova Science Publishers, New York, 2010), 1-7.
A. Bakan, St. Ruscheweyh, L. Salinas, Universally starlike and Pick functions, JAMA, 142 (2020), 539–586. https://doi.org/10.1007/s11854-020-0143-2.
S. Banga, S. Sivaprasad Kumar, The sharp bounds of the second and third Hankel determinants for the class SL∗, Math. Slovaca, 70 (2020), №4, 849–862, https://doi.org/ 10.1515/ms-2017-0398.
T. Hayami, S. Owa, Generalized Hankel determinant for certain classes, Int. J. Math. Anal., 4 (2010), №52, 2573–585.
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), №3, 435–445.
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., 42 (2019), №2, 767–780.
R.J. Libera, E.J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87 (1983), №2, 251–257.
Ch. Pommerenke, Univalent functions, G¨ottingen: Vandenhoeck and Ruprecht, 1975.
Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. Lond. Math. Soc., 41 (1966), 111–122. https://doi.org/10.1112/jlms/s1-41.1.111
B. Rath, K.S. Kumar, D.V. Krishna, A. Lecko, The sharp bound of the third Hankel determinant for starlike functions of order 1/2, Complex Anal. Oper. Theory, (2022), https://doi.org/10.1007/s11785-022-01241-8.
K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan, 11 (1959), 72–75.
M.M. Sheremeta, Yu.S. Trukhan, Starlike and convexity properties for p-valent solutions of the Shah differential equation, Mat. Stud., 48 (2017), №1, 14–23. doi:10.15330/ms.48.1.14-23.
Y.J. Sim, A. Lecko, D.K. Thomas, The second Hankel determinant for strongly convex and Ozaki closeto-convex functions, Annali di Matematica, 200 (2021), 2515–2533. https://doi.org/10.1007/s10231-021-01089-3.
V. Kumar, S. Kumar, V. Ravichandran, Third Hankel Determinant for CertainClasses of Analytic Functions, Mathematical Analysis I: Approximation Theory, February 2020, doi: 10.1007/978-981-15-1153-019.
P. Zaprawa, Third Hankel determinants for subclasses of univalent functions, Mediterr. J. Math., 14 (2017), №1, 1–10.
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