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

For $f\in \mathcal{S}$, the class univalent functions 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 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 $\mathrm{\mid }{a}_4\mathrm{\mid }-\mathrm{\mid }{a}_3\mathrm{\mid }$ when $f\in \mathcal{S}$.