View source for Open mapping theorem (complex analysis)

In [[complex analysis]], the '''open mapping theorem''' states that if <math>U</math> is a [[Domain (mathematical analysis)|domain]] of the [[complex plane]] <math>\mathbb{C}</math> and <math>f: U\to \mathbb{C}</math> is a non-constant [[holomorphic function]], then <math>f</math> is an [[open map]] (i.e. it sends open subsets of <math>U</math> to open subsets of <math>\mathbb{C}</math>, and we have [[invariance of domain]].).

The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the [[real line]], for example, the differentiable function <math>f(x)=x^2</math> is not an open map, as the image of the [[open interval]] <math>(-1, 1)</math> is the half-open interval <math>[0, 1)</math>. 

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 (if constant) or two (if non-constant) but never of dimension 1.

==Proof==

[[Image:Openmappingtheorem.png|thumb|right|upright=1.4| Black dots represent zeros of <math>g(z)</math>. Black annuli represent poles. The boundary of the open set <math>U</math> is given by the dashed line. Note that all poles are exterior to the open set. The smaller red disk is  <math>B</math>, centered at <math>z_0</math>.]]

Assume <math>f: U\to \mathbb{C}</math> is a non-constant holomorphic function and <math>U</math> is a [[Domain (mathematical analysis)|domain]] of the complex plane. We have to show that every [[Point (geometry)|point]] in <math>f(U)</math> is an [[interior point]] of <math>f(U)</math>, i.e. that every point in <math>f(U)</math> has a neighborhood (open disk) which is also in <math>f(U)</math>.

Consider an arbitrary <math>w_0</math> in <math>f(U)</math>. Then there exists a point <math>z_0</math> in <math>U</math> such that <math>w_0 = f(z_0)</math>. Since <math>U</math> is open, we can find <math>d > 0</math> such that the closed disk <math>B</math> around <math>z_0</math> with radius <math>d</math> is fully contained in <math>U</math>. Consider the function <math>g(z)=f(z)-w_0</math>. Note that <math>z_0</math> is a [[root of a function|root]] of the function. 

We know that <math>g(z)</math> is non-constant and holomorphic. The roots of <math>g</math> are isolated by the [[identity theorem]], and by further decreasing the radius of the disk <math>B</math>, we can assure that <math>g(z)</math> has only a single root in <math>B</math> (although this single root may have multiplicity greater than 1).

The boundary of <math>B</math> is a circle and hence a [[compact set]], on which <math>|g(z)|</math> is a positive [[continuous function]], so the [[extreme value theorem]] guarantees the existence of a positive minimum <math>e</math>, that is, <math>e</math> is the minimum of <math>|g(z)|</math> for <math>z</math> on the boundary of <math>B</math> and <math>e>0</math>. 

Denote by <math>D</math> the open disk around <math>w_0</math> with [[radius]] <math>e</math>. By [[Rouché's theorem]], the function <math>g(z)=f(z)-w_0</math> will have the same number of roots (counted with multiplicity) in <math>B</math> as <math>h(z) := f(z)-w_1</math> for any <math>w_1</math>  in <math>D</math>.  This is because 
<math>h(z) = g(z) + (w_0-w_1)</math>, and for <math>z</math> on the boundary of <math>B</math>, <math>|g(z)| \geq e > |w_0-w_1|</math>.  Thus, for every <math>w_1</math> in <math>D</math>, there exists at least one <math>z_1</math> in <math>B</math> such that <math>f(z_1) = w_1</math>. This means that the disk <math>D</math> is contained in <math>f(B)</math>. 

The image of the ball <math>B</math>, <math>f(B)</math> is a subset of the image of <math>U</math>, <math>f(U)</math>. Thus <math>w_0</math> is an interior point of <math>f(U)</math>. Since <math>w_0</math> was arbitrary in <math>f(U)</math> we know that <math>f(U)</math> is open. Since <math>U</math> was arbitrary, the function <math>f</math> is open.

== Applications ==
*[[Maximum modulus principle]]
*[[Rouché's theorem]]
*[[Schwarz lemma]]

== See also ==
* [[Open mapping theorem (functional analysis)]]

== References ==
* {{citation|first=Walter|last=Rudin|authorlink=Walter Rudin|title=Real & Complex Analysis|publisher=McGraw-Hill|year=1966|isbn=0-07-054234-1}}

[[Category:Theorems in complex analysis]]
[[Category:Articles containing proofs]]