Open mapping theorem for holomorphic functions

Statement

If $D\subseteq\mathbb C$ is a domain and $f:D\to\mathbb C$ is nonconstant and holomorphic, then $f$ is an open map: $f(U)$ is open in $\mathbb C$ for every open $U\subseteq D$ .

Local mechanism

Near $a\in D$ , factor

f(z)-f(a)=(z-a)^m g(z),\qquad g(a)\ne0.

Small circles around $a$ therefore wind $m$ times around $f(a)$ , forcing the image of a small neighborhood to contain a neighborhood of $f(a)$ .

Consequences

A bijective holomorphic map between plane domains has holomorphic inverse: continuity of the inverse follows because the map is open, and local complex differentiability follows away from critical points; injectivity rules those out. The theorem also gives a short proof of the maximum modulus principle.

Disambiguation

This result is distinct from the Banach-space open mapping theorem for bounded surjective linear operators. Its hypotheses and proof are specifically complex analytic.

References

John B. Conway, Functions of One Complex Variable I, 2nd ed., Springer, 1978. Publisher record. Relevant: Chapter III, §7.