I'm studying for a complex analysis exam, and I'm stuck on this problem from an old exam:
Let $g$ be a holomorphic function on $|z|<R,R>1$, with $|g(z)|\leq 1$ for all $|z|\leq 1$.
(a) Show that for all $t\in C$ with $|t|<1$, the equation $$z=tg(z)$$ has a unique solution $z=s(t)$ in the disc $|z|<1$.
(b) Show that $t\mapsto s(t)$ is a holomorphic function on the disc $|t|<1$. (Hint: find an integral formula for $s$.)