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 $f(z)=z/(1-z^2).$