A bounded holomorphic function on an unbounded domain is constant
Is this true?
No — it is false.
Others that fail the same way
- Bounded real part ⇒ entire function is constant — fails if only |f| is bounded on a half-plane
- Zeros of a non-constant analytic function can accumulate inside the domain
- |f| bounded near an isolated singularity ⇒ pole
- The Cauchy–Riemann equations at a point imply complex differentiability there
- A harmonic function on any domain has a harmonic conjugate
- Zeros of a non-constant holomorphic function cannot accumulate
From Cauchy Theory › Liouville, Morera, maximum modulus principle