finding all functions analytic with $|f(z)|leq |z+3/2|$ and $f(0)=0$

Let $f$ analytic in $D=left{z:|z|<1right}$ such that $f(0)=0$ and $|f(z)|leq |z+3/2|$ for all $zin D$.
Show that:

(a) $|f(1/2)|leq 1$

(b) Find all functions $f$ that holds (a)

My solution for (a)
Let $g(z)=f(z)/(z+3/2)$. then $g(0)=0$ and $|g(z)|leq 1$. By Lemma’s Schwartz, $|g(z)|leq |z|$ for all $zin D$ then $|f(z)|leq |z|(|z+3/2|)$ therefore $|f(1/2)|leq 1$.

How proves (b)? some hint?