Open mapping theorem (complex analysis): Difference between revisions

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{{Short description|Theorem on holomorphic functions}}

{{Otheruses|Open mapping theorem}}

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''<sup>^2</supmath> is not an open map, as the image of the [[open interval]] <math>(−1-1, 1)</math> is the half-open interval <math>[0, 1)</math>.

In [[complex analysis]], the '''open mapping theorem''' states that if ''U'' is a [[Domain (mathematical analysis)|___domain]] 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''', 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 ''f''(''x'') = ''x''² 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 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.

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==Proof==

[[Image:Openmappingtheorem.png|thumb|right|500pxupright=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 ''z''<submath>0z_0</submath>.]]

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

We know that ''<math>g''(''z'')</math> is not non-constant and holomorphic. The roots of ''<math>g''</math> are isolated by the [[identity theorem]], and by further decreasing the radius of the image disk ''d''<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 ''w''<submath>0w_0</submath> with [[radius]] ''<math>e''</math>. By [[Rouché's theorem]], the function ''<math>g''(''z'') = ''f''(''z'')−''w''<sub>0-w_0</submath> will have the same number of roots (counted with multiplicity) in ''<math>B''</math> as ''<math>h''(''z'') :='' f''(''z'')−''w<sub>1-w_1</submath>'' for any ''w<submath>1w_1</submath>'' in ''<math>D''</math>. This is because

''<math>h''(''z'') = ''g''(''z'') + (''w''₀ w_0- ''w''<sub>1w_1)</submath>), and for ''<math>z''</math> on the boundary of ''<math>B''</math>, <math>|''g''(''z'')| ≥\geq ''e'' > |''w''₀ w_0- ''w''<sub>1w_1|</submath>|. Thus, for every ''w''<submath>1w_1</submath> in ''<math>D''</math>, there exists at least one ''z''<submath>1z_1</submath> in ''<math>B''</math> such that ''f''(''z''<sub>1</submath>f(z_1) = ''w<sub>1w_1</submath>''. 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 ''w''<submath>0w_0</submath> is an interior point of ''<math>f''(''U'')</math>. Since ''w''<submath>0w_0</submath> 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 ==