Open mapping theorem (complex analysis): Difference between revisions

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{{Short description|Theorem on holomorphic functions}}

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>.

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==

[[Image:OpenMapping1Openmappingtheorem.png|framethumb|right|Blueupright=1.4| Black dots represent zeros of ''<math>g''(''z'')</math>. Black spikesannuli 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 circledisk is the set ''<math>B''</math>, constructedcentered in theat proof<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> ishas containeda inneighborhood a(open disk) which is containedalso 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]], andon which <math>|''g''(''z'')|</math> is a positive [[continuous function]], so the [[extreme value theorem]] guarantees the existence of a positive minimum. Let<math>e</math>, ''that is, <math>e''</math> beis the minimum of <math>|''g''(''z'')|</math> for ''<math>z''</math> on the boundary of ''<math>B'',</math> aand positive number<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 ''f''<math>h(''z'')−''w'' for:= any ''w'' within a distance ''e'' of ''w''<sub>0f(z)-w_1</submath>. Thus, for everyany ''w'' in ''D'', there exists at least one ''z''<submath>1w_1</submath> in ''B'' so that ''f''(''z''<submath>1D</submath>). = ''w''. This means that the disk ''D'' is contained in ''f''(''B'').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 ''w''<submath>0w_0</submath> is an interior point of the image of an open set by a holomorphic function ''<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 ==

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[[Category:Theorems in complex analysis]]

[[Category:Articles containing proofs]]

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[[it:Teorema della funzione aperta (analisi complessa)]]

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