The power series $f(z):= \sum_{\nu \geq 0}a_{\nu}z^{\nu}$ converges in a disc $B$ centered at $0$. For every $z\in B$ such that $2z$ is also in $B$, $f$ satisfies $f(2z) = (f(z))^{2}$. Show that if $f(0)\neq 0$, then $f(z)=\mathrm{exp} bz$, with b:=f'(0)=a_{1}.
I'm completely lost with this problem. Could someone give me a hint?