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 $f(z)=z/(1-z^2).$
We present the sharp bounds for the third Hankel determinant [Formula: see text] and Zalcman functional [Formula: see text] of the inverse function of the familiar subfamily of starlike functions associated with the right half of lemniscate of Bernoulli.
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