$\sum_n f_n$ converges uniformly on $|z|\leq r$ for $r<1$ where $f$ entire, $f_n = f\circ \cdots \circ f$, $f(0) = 0$, $|f'(0)| < 1$, $f(D)\subset D$

Question

Let $f(z)$ be an entire function such that $f(0) = 0$, $|f'(0)| < 1$, and $f(D)\subset D$, where $D$ is the open unit disk in $\Bbb C$. Put $f_n = f\circ \cdots \circ f$ for $n=1,2,\dots$, where the composition is done $n$ times. I am asked to prove $\sum_n f_n$ converges uniformly on $|z|\leq r$ for each $r<1$. But I can't see where to use the conditions $f(0)=0$ and $|f'(0)|<1$. Any hints?

Answer 1

For $0 < r < 1$ we have $$ \max \left\{ \frac{|f(z)|}{|z|} : |z| = r \right \} \le 1 $$ according to the Schwarz Lemma. Equality cannot hold because $f$ is not a rotation by a factor of modulus one.

It follows that there is a constant $c < 1$ such that $$ |f(z)| \le c |z| $$ for all $|z| \le r$, so that $$ |f_n(z)| \le c^n |z| < c^n $$ uniformly on $|z| \le r$.