= Solution
Because $f(0)=0$, the <Taylor series> of $f$ at zero shows that
$$
h(z)=\begin{cases}f(z)/z,&z\ne0,\\f'(0),&z=0\end{cases}
$$
is <holomorphic> throughout the <unit disk>. Fix $0<r<1$. On $|z|=r$ we have $|h(z)|<1/r$, and the <maximum modulus principle> on the closed disk of radius $r$ gives $|h(z)|\leq1/r$ for $|z|\leq r$. For any fixed $z$ inside the <unit disk> we may let $r$ increase to one, obtaining $|h(z)|\leq1$. Hence the <Schwarz lemma> inequalities are
$$
\boxed{|f(z)|\leq|z|,\qquad |f'(0)|\leq1.}
$$
The <derivative> bound is the same estimate at $z=0$, using the holomorphic extension of the quotient.
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