Let $\mathbb{D}$ be the unit disk in $\mathbb{C}$, let $f:\mathbb{D}\to \mathbb{D}$ be holomorphic. Then $|f^{\prime }(0)|\le$ 1;;1 and $|f(z)|\le$ 1;; $|z|$ for all $z\in \mathbb{D}$. Furthermore, if either of the inequalities are equalities (the latter for at least one point of $\mathbb{D}$), then ==1;;$f(z)=e^{i\theta }z$.