Open mapping theorem (complex analysis): Difference between revisions

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In [[complex analysis]], the '''open mapping theorem''' states that if ''U'' is an 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'') = ''x''² is not an open map, as the image of the [[open interval]] (−1−1, 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.

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[[Image:OpenMapping1.png|frame|right|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 <math>''U</math>'' is a [[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 ''w''<mathsub>w_00</mathsub> in <math>''f''(''U'')</math>. Then there exists a point ''z''<mathsub>z_00</mathsub> in ''U'' such that ''w''<mathsub>0</sub>w_0 = ''f''(z_0)''z''<sub>0</mathsub>). Since ''U'' is open, we can find <math>''d'' > 0</math> such that the closed disk <math>''B</math>'' around ''z''<mathsub>z_00</mathsub> with radius ''d'' is fully contained in ''U''. Consider the function <math>''g''(''z'') = ''f''(''z'')-w_0−''w''<sub>0</mathsub>. Note that ''z''<mathsub>z_00</mathsub> is a [[root of a function|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).

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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 <math>''D</math>'' the disk around ''w''<mathsub>w_00</mathsub> with [[radius]] ''e''. By [[Rouché's theorem]], the function <math>''g''(''z'') = ''f''(''z'')-w_0−''w''<sub>0</mathsub> will have the same number of roots (counted with multiplicity) in ''B'' as <math>''f''(''z'')-−''w</math>'' for any <math>''w</math>'' within a distance <math>''e</math>'' of ''w''<mathsub>w_00</mathsub>. Thus, for every <math>''w</math>'' in <math>''D</math>'', there exists at least one ''z''<mathsub>z_11</mathsub> in <math>''B</math>'' so that <math>''f''(z_1''z''₁) = ''w</math>''. This means that the disk ''D'' is contained in <math>''f''(''B'')</math>.

The image of the ball ''B'', <math>''f''(''B'')</math> is a subset of the image of ''U'', <math>''f''(''U'')</math>. Thus ''w''<mathsub>w_00</mathsub> is an interior point of the image of an open set by a holomorphic function <math>''f''(''U'')</math>. Since ''w''<mathsub>w_00</mathsub> 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 ==