If $f$ analytic in $G$ and $\exists \lambda >0$, $|f(z)| = \lambda, \forall z \in \partial G \Rightarrow \exists z, f(z) = 0$

Question

Let $f$ be a non-constant analytic function in $G\subseteq \mathbb{C}$ (connected, bounded and open) and cont. in $\overline{G}$ such that

$$\exists \lambda >0, \text{ with } |f(z)| = \lambda, \forall z \in \partial G$$

Then $$\exists z \in G, \text{ with } f(z) = 0$$

I have two different approaches designed by me and I want to see if I got both right.

First:

Since $f$ is analytic in $G$ and continuous in $\overline{G}$ by the Maximum Modulus Theorem (Second version of Conway's book)

$$\max_{z \in \overline{G}}\{|f(z)|\} = \max_{z \in \partial G}\{|f(z)|\}$$

Then we have that $|f(z)| \leq \lambda, \forall z \in \overline{G}$.

Now consider the function

$$g(z) = \frac{1}{f(z)} $$

Assume $f(z) = 0$ has no solutions in $G$, then $g(z)$ is analytic in $G$ and continuous in $\overline{G}$ because $f$ is, also $g$ satisfies

$$|g(z)| = \frac{1}{|f(z)|} \leq \frac{1}{\lambda}, \forall z \in \overline{G}$$

By the Maximum modulus theorem (first version of Conway's book), there exists $z_m \in \overline{G}$ such that $|g(z_m)| = \frac{1}{\lambda} \geq g(z)$ for all $z \in \overline{G}$. Thus, $g$ is constant which implies that $f$ is constant and we get a contradiction.

Second proof:

Since $f$ is analytic in $G$ and continuous in $\overline{G}$ by the Maximum Modulus Theorem (Second version of Conway's book)

$$\max_{z \in \overline{G}}\{|f(z)|\} = \max_{z \in \partial G}\{|f(z)|\}$$

Also, $f$ is non-constant and by the Open Mapping Theorem, $f$ must map open sets to open sets, but the modulus of $f$ is bounded for every point in $G$, then $f(G) \subset \{z \in \mathbb{C} : |z| < \lambda\}$ and because $f$ is continuous in $\overline{G}$, then $f(\partial G)$ is the border of the disk or contained in it.Hence, the $\operatorname{Range}(f) = \{z \in \mathbb{C}: |z| \leq 1 \}$, and disk pass through the origin, and must exist a point in $G$ such that f(z) = 0.

I will want to make a better description of the second one, but I think that it is enough. Which one is better for a qualifying exam and if I have to fix some details please let me know.

Answer 1

First version is not accurate, specifically

... there exists $z_m\in \overline{D}$ such that $|g(z_m)|=\frac{1}{λ}\geq g(z)$ for all $z∈\overline{D}$. Thus, $g$ is constant ...

$\overline{D}$ is closed, MMP restricts the result to open sets, consequence of this is that $g$ reaches the maximum at the boundary $\partial G \subset \overline{D}$. So $z_m \in \partial G \subset \overline{D}$, no contradictions so far.

To fix the problem, you need to fix a point $z_0 \in D$ (open set!) for which $|f(z_0)|<\lambda$ (you already deducted this), thus $|g(z_0)|=\left|\frac{1}{f(z_0)}\right| > \frac{1}{\lambda} \geq |g(z)|$. So $\exists z_0 \in D$ such that $|g(z_0)|\geq |g(z)|, \forall z \in D$ and only now you can conclude that $g$ is constant in $D$.