On the difference of coefficients of univalent functions

Abstract

For $f\in \mathcal{S}$, the class of normalized functions, analytic and univalent 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 give an upper bound for the coefficient difference $|a_4|-|a_3|$ when $f\in \mathcal{S}$. This provides an improved bound in the case $n=3$ of Grispan's 1976 general bound $||a_{n+1}|-|a_n||\le 3.61\dots .$ Other coefficients bounds, and bounds for the second and third Hankel determinants when $f\in \mathcal{S}$ are found when either $a_2=0,$ or $a_3=0$.