Suppose $f$ is a holomorphic function (not necessarily bounded) on $\mathbb{D}$ such that $f(0) = 0$. Prove the the infinite series $\sum_{n=1}^\infty f(z^n)$ converges uniformly on compact subsets of $\mathbb{D}$.
I met this problem on today's qual. Here is what I have so far, since $f(0) = 0$, we can write $f(z) = z^m h(z)$ for some integer $m$. Then $f(z^n) = z^{nm}h(z^n)$. We might then use Cauchy's criterion for uniform convergence to finish the proof.