Open mapping theorem (complex analysis)

In complex analysis, the open mapping theorem states that if U is a open subset of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map (i.e. it sends open subsets of U to open subsets of C).

The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the real line, for example, the differentiable function f(x) = x² is not an open map, as the image of the open interval (−1,1) is the half-open interval [0,1).

The theorem for example implies that a non-constant holomorphic function cannot map an open disk onto a portion of any real line embedded in the complex plane. Images of holomorphic functions can be of real dimension zero (if constant) or two (if non-constant) but never of dimension 1.

Proof

Blue dots represent zeros of g(z). Black spikes represent poles. The boundary of the open set U is given by the dashed line. Note that all poles are exterior to the open set. The smaller red circle is the set B constructed in the proof.

Assume f:U → C is a non-constant holomorphic function and $U$ is a open subset of the complex plane. We have to show that every point in $f(U)$ is an interior point of $f(U)$, i.e. that every point in $f(U)$ is contained in a disk which is contained in $f(U)$.

Consider an arbitrary $w_{0}$ in $f(U)$. Then there exists a point $z_{0}$ in U such that $w_{0}=f(z_{0})$. Since U is open, we can find $d>0$ such that the closed disk $B$ around $z_{0}$ with radius d is fully contained in U. Consider the function $g(z)=f(z)-w_{0}$. Note that $z_{0}$ is a root of the function.

We know that g(z) is not constant and holomorphic. The roots of g are isolated and by further decreasing the radius of the image disk d, we can assure that g(z) has only a single root in B (although this single root may have multiplicity greater than 1).

The boundary of B is a circle and hence a compact set, and |g(z)| is a continuous function, so the extreme value theorem guarantees the existence of a minimum. Let e be the minimum of |g(z)| for z on the boundary of B, a positive number.

Denote by $D$ the disk around $w_{0}$ with radius e. By Rouché's theorem, the function $g(z)=f(z)-w_{0}$ will have the same number of roots (counted with multiplicity) in B as $f(z)-w$ for any $w$ within a distance $e$ of $w_{0}$. Thus, for every $w$ in $D$, there exists at least one $z_{1}$ in $B$ so that $f(z_{1})=w$. This means that the disk D is contained in $f(B)$.

The image of the ball B, $f(B)$ is a subset of the image of U, $f(U)$. Thus $w_{0}$ is an interior point of the image of an open set by a holomorphic function $f(U)$. Since $w_{0}$ was arbitrary in $f(U)$ we know that $f(U)$ is open. Since U was arbitrary, the function $f$ is open.

Applications

Maximum modulus principle

References

• Rudin, Walter (1966), Real & Complex Analysis, McGraw-Hill, ISBN 0-07-054234-1