Open mapping theorem (complex analysis): Difference between revisions

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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 real line embedded in the complex plane. Images of holomorphic functions can be of dimension zero (if constant) or two (if non-constant) but never of dimension 1.

==Proof==

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Assume ''f'':''U'' &rarr; '''C''' is a non-constant holomorphic function and <math>U</math> is a connected [[open subset]] 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> is contained in a disk which is contained 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 ''U'' such that <math>w_0 = f(z_0)</math>. Since ''U'' is open, we can find <math>d>0</math> such that the closed disk <math>B</math> around <math>z_0</math> with radius ''d'' is fully contained in ''U''. Since ''U'' is connected and ''f'' is not constant on ''U'', we then know that ''f'' is not constant on ''B'' because of [[Analytic continuation]]. 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 ''g''(''z'') is not constant, and byholomorphic. furtherThe decreasingreciprocal ''d'',of weany can assure thatholomorphic ''g''(''z'') hasis only[[meromorphic]] aand singlehas rootisolated in ''B''poles. (TheThus the roots of holomorphic non-constant functions are isolated.) LetIn particular, the roots of ''eg'' beare isolated and by further decreasing the minimumradius of |the image disk ''d'', we can assure that ''g''(''z'')| forhas ''z''only ona thesingle boundaryroot ofin ''B'', a positive number. (

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 thisa minimum. Let ''e'' be the minimum of |''g''(''z'')| for ''z'' on the boundary of ''B'', a positive number.

Denote by <math>D</math> the 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 in ''B'' as <math>f(z)-w</math> for any <math>w</math> within a distance <math>e</math> of <math>w_0</math>. Thus, for every <math>w</math> in <math>D</math>, there exists one (and only one) <math>z_1</math> in <math>B</math> so that <math>f(z_1) = w</math>. This means that the disk ''D'' is contained in <math>f(B)</math>, which is a subset of <math>f(U)</math>. Thus <math>w_0</math> is an [[interior point]] of <math>f(U)</math>.

The image of the ball ''B'', <math>f(B)</math> is a subset of the image of ''U'', <math>f(U)</math>. Thus <math>w_0</math> is an [[interior point]] of the image of an open set by a holomorphic function <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 ''U'' was arbitrary, the function <math>f</math> is open.

== Applications ==

[[Maximum modulus principle]]

== References ==