  Let ${\mathcal A}$ denote the family of all functions $f$ analytic in the
unit disk $\ID$ and satisfying the normalization $f(0)=0= f'(0)-1$. Let
$\mathcal{S}$ denote the subclass of ${\mathcal A}$ consisting of univalent
functions in $\ID$. We consider the subclass $\mathcal{U} $ of $\mathcal{S}$
that is defined by the condition that for its members $f$ the condition $$\left
|\left (\frac{z}{f(z)} \right )^{2}f'(z)-1\right | < 1 ~\mbox{ for $z\in \ID$}
$$ holds. To theses relations belong striking similarities and on the other
hand big differences. We show that some results about $\mathcal{S}$ can be
improved for $\mathcal{U}$, while others cannot.
