Open mapping theorem (complex analysis)


In complex analysis, the open mapping theorem states that if U is a domain of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map.
The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the real line, for example, the differentiable function f = x2 is not an open map, as the image of the open interval.
The theorem for example implies that a non-constant holomorphic function cannot map an open disk onto a portion of any line embedded in the complex plane. Images of holomorphic functions can be of real dimension zero or two but never of dimension 1.

Proof

Assume f : U → C is a non-constant holomorphic function and U is a domain of the complex plane. We have to show that every point in f is an interior point of f, i.e. that every point in f has a neighborhood which is also in f.
Consider an arbitrary w0 in f. Then there exists a point z0 in U such that w0 = f. Since U is open, we can find d > 0 such that the closed disk B around z0 with radius d is fully contained in U. Consider the function g = f−w0. Note that z0 is a root of the function.
We know that g is not constant and holomorphic. The roots of g are isolated by the identity theorem, and by further decreasing the radius of the image disk d, we can assure that g has only a single root in B.
The boundary of B is a circle and hence a compact set, on which |g| is a positive continuous function, so the extreme value theorem guarantees the existence of a positive minimum e, that is, e is the minimum of |g| for z on the boundary of B and e > 0.
Denote by D the open disk around w0 with radius e. By Rouché's theorem, the function g = f−w0 will have the same number of roots in B as h:=f−w1 for any w1 in D. This is because
h = g +, and for z on the boundary of B, |g| ≥ e > |w0 - w1|. Thus, for every w1 in D, there exists at least one z1 in B such that f = w1. This means that the disk D is contained in f.
The image of the ball B, f is a subset of the image of U, f. Thus w0 is an interior point of f. Since w0 was arbitrary in f we know that f is open. Since U was arbitrary, the function f is open.

Applications