Is the maximum modulus of an entire function on a circle of radius $r$ a smooth function of $r$?

Question

Let $f$ be a non-constant holomorphic function on $\mathbb{C},$ for any $r>0,$ set $$M(r)=\max_{|z|=r}|f(z)|$$ Is $M(r)\in C^{\infty}(0,+\infty)$ ? I have showed that $M(r)$ is continuous and strictly increasing, but I have no idea to deal with the derivative of $M(r),$ can someone help me?

Answer 1

Take $f(z)=(1-z)e^z$ then if $r\le 2$ we have $r=2\sin \theta/2$ for a unique $0\le \theta \le \pi$ and it is a good exercise to show that if $z_r=1-e^{i\theta}$ has modulus as above, $f$ attains maximum modulus at $z_r$ which gives $M(f,r)=e^{r^2/2}, r\le 2$; however if $r\ge 2$ it is not hard to show that $M(f,r)=(r-1)e^r$ obtained at $z=r$ so $M$ is not smooth at $2$.