  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 $a_3=0$.
