Check $r\mapsto M_r=\max\{|f(z)|~:~|z|=r\}$ is an increasing function

Question

Let $f$ be a non constant entire function and $M_r=\max\{|f(z)|~:~|z|=r\}$. I need to check whether the mapping $r\mapsto M_r$, is increasing.

I am trying to prove it by the open mapping theorem, but that is not sufficient. Please help.

Answer 1

Maximum Modulus Theorem tells you that $M_r = \max \{|f(z)|:|z| \leq r\}$. It is now obvious that $M_r$ is increasing.