Maximum modulus theorem for holomorphic function on an unbounded region

Question

Suppose $\Gamma$ is the boundary of an unbounded region $\Omega$, $f \in H(\Omega)$, $f$ is continuous on $\Omega \cup \Gamma$, and there are constants $B < \infty$ and $M < \infty$ such that $ |f| \leq M$ on $\Gamma$ and $ |f| \leq B $ in $\Omega$. Prove that we then actually have $ |f| \leq M $ in $\Omega$.

Proof:

According to the suggestion,
fix $z_0 \in \Omega$ and let n be a large integer. Let $V$ be a large disc with center at $0$, and apply the maximum modulus theorem to the function $f(z)^n/z$ in the component of $V \cap \Omega$ which contains $z_0$.

Then
$$
\left|\frac{f(z_0)^n}{z_0}\right| \leq \max\left(\frac{B^n}{r}, M^n\right)
$$
r is the radius of $V$. If $r$ is large enough this means $|\frac{f(z_0)^n}{z_0}| \leq M^n$, for $n \to \infty$ it means $|f(z_0)|<M$, as $z_0$ is arbitrary $ \in \Omega$, this means $|f(z)| \leq M$.

Is my proof correct?

Answer 1

There is a problem with your proof: The function $f(z)^n/z$ is not holomorphic in $\Omega$ if $0 \in \Omega$, and it is not bounded on the boundary of $V \cup \Omega$ if $0 \in \Gamma$.

If $0 \in \Bbb C \setminus \overline \Omega$ then $\delta = \operatorname{dist}(0, \Gamma) > 0$, and your approach works with the estimate $$ \left|\frac{f(z_0)^n}{z_0}\right| \leq \max\left(\frac{B^n}{r}, \frac{M^n}{\delta}\right) \, . $$ If there is any other point $z_1$ in $\Bbb C \setminus \overline \Omega$ then the same idea works by considering $f(z)^n/(z-z_1)$ instead.

But all this fails if $\overline \Omega = \Bbb C$, i.e. if there is no open disk in the complement of the domain. In that case one has to replace $1/z$ by a different function, see below.

Also note that the statement is wrong if the boundary of $\Omega$ is empty, i.e. if $\Omega = \Bbb C$. Then the condition “$|f|\le M$ on $\Gamma$” holds for arbitrary values of $M$ (as a “vacuous truth”), but that does not imply that $|f| \le M$ in $\Omega$.

A correct formulation of the statement would be:

Let $\Omega \subset \Bbb C$ be a domain with non-empty boundary $\Gamma$, and $f: \Omega \cup \Gamma \to \Bbb C $ a continuous function which is holomorphic in $\Omega$. Suppose there are real constants $M$ and $B$ such that

• $|f(z)| \le M$ for all $z \in \Gamma$, and
• $|f(z)| \le B$ for all $z \in \Omega$.

Then $|f(z)| \le M$ for all $z \in \Omega$.

(The following proof is taken from Bak J., Newman D.J. (2010) Maximum-Modulus Theorems for Unbounded Domains. In: Complex Analysis. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7288-0_15 .)

We may assume that $f$ is not constant, because the statement is trivially true for constant functions. (Why?)

Fix $z_0 \in \Bbb \Omega$ and a positive integer $n$. We choose any $z_1 \in \Omega$ with $f(z_1) \ne f(z_0)$ and define the function $g$ on $\Omega \cup \Gamma$ as $$ g(z) = \begin{cases} \frac{f(z)-f(z_1)}{z-z_1} & \text{ if } z \ne z_1 \, , \\ f'(z_1) & \text{ if } z = z_1 \, . \end{cases} $$

$g$ replaces the function $1/z$ from your proof. It has the following properties:

• $g$ is holomorphic in $\Omega$ and continuous on $\Omega \cup \Gamma$.
• $\lim_{z \to \infty} g(z) = 0$ (because $f$ is bounded).
• $g$ is bounded on $\Omega \cup \Gamma$.

We set $K = \sup \{z \in \Omega \cup \Gamma : |g(z)| \}$ and choose $r> 0$ so large such that

• $|z_0| < r$, and
• $|g(z)| < K M^n/B^n$ for all $z \in \Omega$ with $|z| = r$.

Now we can apply the maximum modulus theorem to $h(z) = f(z)^n g(z)$ on $\Omega \cap B_r(0)$. On boundary points $z$ with $|z| = r$ we have $$ |f(z)^n g(z)| \le B^n |g(z)| \le M^n K $$ and on boundary points $z \in \Gamma$ $$ |f(z)^n g(z)| \le M^n K $$ holds as well. It follows that $$ |f(z_0)| \le M \left( \frac{K}{|g(z_0)|}\right)^{1/n} \, , $$ and for $n \to \infty$ we conclude that $|f(z_0)| \le M$.

Question to the reader: Where does the proof use that the boundary of the domain is not empty?