I ran into this reading some Complex stuff for fluid dynamics, and it seems so simple but it's got me stuck. $D$ stands for the unit disk.
Since $\mid z \mid<1$, then $\mid z^2\mid \leq \mid z \mid$. So $$\mid f(z^2)\mid\geq \mid f(z)\mid$$ is trying to tell me that points closer to the origin are being sent further out, and geometrically it feels like a holomorphic function that is not constant shouldn't be able to do that, but I don't quite see why.
I know the Maximum Modulus principle is usually the way to get constants out of holomorphic maps. Also, it must be easy to show that $\mid f(z) \mid \leq 1$ and $f(0)=0$, so that I also have the Schwarz Lemma at hand to give me $\mid f(z) \mid \leq \mid z\mid$. Playing with this, however, I've only managed to get the fact that $$\mid f(z)\mid \leq \mid f(z^2)\mid\leq \mid z^2 \mid\leq \mid z\mid$$ Would this be helpful in finding a maximum inside the unit disk to use Max/Mod to get a constant out of $f$?
Do you have any other suggestions? Thanks!
Hint: The hypotheses imply $\sup_{|z|\le 1/2}\,|f| \le \sup_{|z|\le 1/4}\,|f|.$