Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Liouville's Theorem states that if a function is bounded and holomorphic on the complex plane (i.e. bounded and entire), then it is a constant function.

What if we consider the following, slightly modified scenario:

Suppose a function $f$ is holomorphic and has constant modulus on a bounded domain $D$ (e.g. a small disk).

Can we use Liouville's Theorem to somehow conclude that $f$ is a constant function? (either on $D$ or on the whole of the complex plane?)

share|cite|improve this question
Maybe not Liouvill, but maximum modulus principle will work. – Davide Giraudo Dec 5 '12 at 21:27
Thanks Davide - indeed, this makes it much simpler. – Conan Wong Dec 5 '12 at 21:33
@DavideGiraudo, can you say more on how to use the maximum modulus principle here? – user27126 Dec 5 '12 at 22:13
A constant modulus on the closure of a domain gives that the function is constant. – Davide Giraudo Dec 5 '12 at 22:22
up vote 2 down vote accepted

I don't see how you could use Liouville's theorem to prove that, but it does follow from Cauchy-Riemann's equations.

If you assume that $f$ is entire, use Cauchy-Riemann's equation on $|f|^2 = u^2 + v^2$ to show that both $u$ and $v$ must be constant on $D$. After that it follows from the uniqueness theorem that $f$ is constant everywhere.

share|cite|improve this answer
Thanks mrf. Just to clarify - instead of "f is entire," the assumption (as in the original scenario) that f is holomorphic on $D$ will suffice right? – Conan Wong Dec 5 '12 at 21:34
Yes. (But of course then you only conclude that $f$ is constant on $D$.) – mrf Dec 5 '12 at 21:35

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.