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|
(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 $s$.)