An exact estimate of the third Hankel determinants for functions inverse to convex functions

Abstract

Invesigation of bounds for Hankel determinat of analytic univalent functions is prominent intrest of many researcher from early twenth century to study geometric properties. Many authors obtained non sharp upper bound of third Hankel determinat for different subclasses of analytic univalent functions until Kwon et al. obtained exact estimation of the fourth coefficeient of Caratheodory class.
Recently authors made use of an exact estimation of the fourth coefficient, well known second and third coefficient of Caratheodory class obtained sharp bound for the third Hankel determinant associated with subclasses of analytic univalent functions.

Let $w=f(z)=z+a_{2}z^{2}+\cdots$ be analytic in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$, and $\mathcal{S}$ be the subclass of normalized univalent functions with $f(0)=0$, and $f'(0)=1$. Let $z=f^{-1}$ be the inverse function of $f$, given by $f^{-1}(w)=w+t_2w^2+\cdots$ for some $|w|<r_o(f)$. Let  $\mathcal{S}^c\subset\mathcal{S}$ be the subset of convex functions in $\mathbb{D}$. In this paper, we estimate the best possible upper bound for the third Hankel determinant for the inverse function $z=f^{-1}$ when $f\in \mathcal{S}^c$.

Let $\mathcal{S}^c$ be the class of convex functions. We prove the following statements (Theorem):
If $f\in$ $\mathcal{S}^c$, then
\begin{equation*}
\big|H_{3,1}(f^{-1})\big| \leq \frac{1}{36}
\end{equation*}
and the inequality is attained for $p_0(z)=(1+z^3)/(1-z^3).$