On the difference of coefficients of univalent functions

For $f\in \mathcal{S}$, the class of normalized functions, analytic and univalent in the unit disk $\mathbb{D}$ and given by $f(z)=z+{\sum }_{n=2}^{\mathrm{\infty }}{a}_n{z}^n$ for $z\in \mathbb{D}$, we give an upper bound for the coefficient difference $\mathrm{\mid }{a}_4\mathrm{\mid }-\mathrm{\mid }{a}_3\mathrm{\mid }$ when $f\in \mathcal{S}$. This provides an improved bound in the case $n=3$ of Grispan’s 1976 general bound $\mathrm{\mid }\mathrm{\mid }{a}_{n+1}\mathrm{\mid }-\mathrm{\mid }{a}_n\mathrm{\mid }\mathrm{\mid }\le 3.61\dots \mathrm{.}$ 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 $a_3=0$.