Show that if $f$ is entire and $(\forall z \in \mathbb{C})$ $|f(z^2)| \leq |f(z)|$, then $f$ is constant

Question

Show that, if $f$ is entire and $(\forall z \in \mathbb{C})$ $|f(z^2)| \leq |f(z)|$, then $f$ is constant

How should I approach this problem? Should I use the power series of $f(z)$ or Liouville's Theorem?

Answer 1

Consider the region $|z|=R$ for $R>1$. Since $\left|f(z^2)\right|\leq\left|f(z)\right|$,

$$ \max_{|z|=R^2}\left|f(z)\right| \leq \max_{|z|=R}\left|f(z)\right|$$ which contradicts the maximum modulus principle over the region $|z|\leq R^2$, unless $f$ is constant.