The sharp bound of the third order Hankel determinant for inverse of Ozaki close-to-convex functions

TL;DR

This paper establishes the precise upper bounds for the third-order Hankel determinant of inverse functions within the Ozaki close-to-convex class, advancing understanding of their geometric function properties.

Contribution

It provides the first sharp bounds for the third-order Hankel determinant specifically for inverse functions in the Ozaki close-to-convex class.

Findings

Sharp bounds for the third-order Hankel determinant are derived.

Results improve the understanding of inverse function behavior in close-to-convex classes.

The bounds are proven to be optimal.

Abstract

Let $f$ be analytic in the unit disk $\mathbb{D}= \{z \in \mathbb{C}~:~ |z| < 1\}$, and $\mathcal{S}$ be the subclass of normalized univalent functions given by $f(z)=\sum_{n=1}^{\infty}a_{n}z^{n},~a_{1}:=1$ for $z \in\mathbb{D}$. We present the sharp bounds of the third-order Hankel determinant for inverse functions when it belongs to of the class of Ozaki close-to-convex.