Improvements of certain results of the class $\mathcal{S}$ of univalent functions

TL;DR

This paper refines bounds on Hankel determinants and coefficient differences for univalent functions in the class , providing sharper estimates under specific coefficient conditions.

Contribution

It introduces improved bounds for second and third Hankel determinants and the coefficient difference  -  for functions in , especially when certain coefficients vanish.

Findings

Sharper bounds for second and third Hankel determinants.

Improved upper bound for |a| - |a|.

Results apply to functions with specific zero coefficients.

Abstract

For $f\in \mathcal{S}$, the class univalent functions in the unit disk $\mathbb{D}$ and given by $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$ for $z\in \mathbb{D}$, we improve previous bounds for the second and third Hankel determinants in case when either $a_2=0,$ or $a_3=0$. We also improve an upper bound for the coefficient difference $|a_4|-|a_3|$ when $f\in \mathcal{S}$.