On $H_3(1)$ Hankel determinant for some classes of univalent functions

TL;DR

This paper determines sharp upper bounds for the Hankel determinant $H_3(1)$ across classes of bounded-turning, starlike, and convex functions in the unit disk, advancing understanding of their coefficient constraints.

Contribution

It provides the first complete sharp bounds for $H_3(1)$ for these specific classes of univalent functions, filling a gap in the existing literature.

Findings

Sharp upper bounds for $H_3(1)$ established for each class.

Results unify and extend previous partial bounds.

Contributes to the coefficient theory of univalent functions.

Abstract

Focus in this paper is on the Hankel determinant, $H_3(1)$, for the well-known classes of bounded-turning, starlike and convex functions in the open unit disk $E=\{z\in \mathbb{C}\colon|z|<1\}$. The results obtained complete the series of research works in the search for sharp upper bounds on $H_3(1)$ for each of these classes.