holomorphic function with nonvanishing derivative on unit disk $D$

Question

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?

Answer 1

Yes: This is true for functions which are smooth enough, and in fact (if $f$ is smooth) we have

$$\int |\cos kx| f(x) dx \to A \int f(x) dx$$

where $A$ is the average of $|\cos x|$ over a single cycle.

Now given $f \in L^1$, choose a sequence of smooth enough $f_n$ converging to $f$ in the $L^1$ norm, and then note

$$\left|\int |\cos kx| f(x) dx - \int |\cos kx| f_n(x) dx\right| \le \int |f - f_n| \to 0$$