How do I have to show that if $f$ is an entire function satisfying $|f(z)|\leq |f(z^2)|$ for all $z\in \Bbb C$ then $f$ is constant?
I know that to show an entire function is constant, Liouville's theorem, or Cauchy's inequality(in order to show that $f'=0$) is useful, but in this question I don't see what theorem should I have to use.
Thanks in advance.
Best Answer
Let $|z|<1$. Then, as suggested by @Conrad in the comments:
$$|f(z)|\le|f(z^2)|\le \lim_{n\to \infty}|f(z^{2^N})|=|f(0)|$$
Thus $0$ is a maximum for the modulus of the function inside the unitary disk. Thanks to the maximum modulus principle, the function is constant.
The result is extended to $\mathbb{C}$ thanks to the identity theorem.