# .

# Open mapping theorem (complex analysis)

In complex analysis, the **open mapping theorem** states that if *U* is a 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.

Proof

Assume *f* : *U* → **C** is a non-constant holomorphic function and *U* is a domain of the complex plane. We have to show that every point in *f*(*U*) is an interior point of *f*(*U*), i.e. that every point in *f*(*U*) has a neighborhood (open disk) which is also in *f*(*U*).

Consider an arbitrary *w*_{0} in *f*(*U*). Then there exists a point *z*_{0} in *U* such that *w*_{0} = *f*(*z*_{0}). Since *U* is open, we can find *d* > 0 such that the closed disk *B* around *z*_{0} with radius *d* is fully contained in *U*. Consider the function *g*(*z*) = *f*(*z*)−*w*_{0}. Note that *z*_{0} is a root of the function.

We know that *g*(*z*) is not constant and holomorphic. The roots of *g* are isolated by the identity theorem, 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).

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

Denote by *D* the open disk around *w*_{0} with radius *e*. By Rouché's theorem, the function *g*(*z*) = *f*(*z*)−*w*_{0} will have the same number of roots (counted with multiplicity) in *B* as *h*(*z*):=*f*(*z*)−*w _{1}* for any

*w*in

_{1}*D*. This is because

*h*(

*z*) =

*g*(

*z*) + (

*w*

_{0}-

*w*

_{1}), and for

*z*on the boundary of

*B*, |

*g*(

*z*)| ≥

*e*> |

*w*

_{0}-

*w*

_{1}|. Thus, for every

*w*

_{1}in

*D*, there exists at least one

*z*

_{1}in

*B*such that

*f*(

*z*

_{1}) =

*w*. This means that the disk

_{1}*D*is contained in

*f*(

*B*).

The image of the ball *B*, *f*(*B*) is a subset of the image of *U*, *f*(*U*). Thus *w*_{0} is an interior point of *f*(*U*). Since *w*_{0} was arbitrary in *f*(*U*) we know that *f*(*U*) is open. Since *U* was arbitrary, the function *f* is open.

Applications

Maximum modulus principle

See also

Open mapping theorem (functional analysis)

References

Rudin, Walter (1966), Real & Complex Analysis, McGraw-Hill, ISBN 0-07-054234-1

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