let $f$ to be an analytic function on $D=\{z\in \mathbb{C}: |z|<1\}$.
Show that there exists an $\epsilon\in (0,1)$ such that for any natural number $m$, there is an analytic function $g=g_m$ on $D_{\epsilon}=\{z\in\mathbb{C}: |z|<\epsilon\}$ for which we have $f(z^m)=(g(z))^m$ holds for every $z\in D_{\epsilon}$.
Any hint on this problem? Thanks in advance.