Maximum and minimum for $f(z)=z^m \Pi_n (1-z/a_n)$ and $g(z)=z^m \Pi_n(1-z/|a_n|)$

Question

Let me use the following notation:

Notation. For an entire function $f$ and $r>0$, let $M(r,f)$ and $m(r,f)$ denote $\max_{|z|=r}|f(z)|$ and $\min_{|z|=r}|f(z)|$, respectively.

Problem.(Ahlfors, Chap.5, Sec. 3.2. #2) Assume that an entire function $f(z)$ has genus zero so that $f(z)=z^m \Pi_n (1-z/a_n)$. Compare $f(z)$ with $g(z)=z^m \Pi_n(1-z/|a_n|)$ and show that $M(r,f) \leq M(r,g)$ and that $m(r,f) \geq m(r,g)$ for all $r>0$.

How do I have to start? I really have no idea. Any help will be greatly appreciated.

Answer 1

For $M(r,f) \leq M(r,g)$, we can use the triangle inequality on each term: $$|1-\frac{z}{a_n}| \leq 1+ |\frac{z}{a_n}|=1+\frac{r}{|a_n|}$$ If we take $z=-r$, we see that $M(r,f) \leq g(-r) \leq M(r,g)$. Can you think of an argument for $m(r,f) \geq m(r,g)$?