Open mapping theorem (complex analysis)

In complex analysis, the open mapping theorem states that if ${\displaystyle U}${\displaystyle U} is a domain of the complex plane ${\displaystyle \mathbb {C} }${\displaystyle \mathbb {C} } and ${\displaystyle f:U\to \mathbb {C} }${\displaystyle f:U\to \mathbb {C} } is a non-constant holomorphic function, then ${\displaystyle f}${\displaystyle f} is an open map (i.e. it sends open subsets of ${\displaystyle U}${\displaystyle U} to open subsets of ${\displaystyle \mathbb {C} }${\displaystyle \mathbb {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 ${\displaystyle f(x)=x^{2}}${\displaystyle f(x)=x^{2}} is not an open map, as the image of the open interval ${\displaystyle (-1,1)}${\displaystyle (-1,1)} is the half-open interval ${\displaystyle [0,1)}${\displaystyle [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.

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

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

We know that ${\displaystyle g(z)}${\displaystyle g(z)} is non-constant and holomorphic. The roots of ${\displaystyle g}${\displaystyle g} are isolated by the identity theorem, and by further decreasing the radius of the disk ${\displaystyle B}${\displaystyle B}, we can assure that ${\displaystyle g(z)}${\displaystyle g(z)} has only a single root in ${\displaystyle B}${\displaystyle B} (although this single root may have multiplicity greater than 1).

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

Denote by ${\displaystyle D}${\displaystyle D} the open disk around ${\displaystyle w_{0}}${\displaystyle w_{0}} with radius ${\displaystyle e}${\displaystyle e}. By Rouché's theorem, the function ${\displaystyle g(z)=f(z)-w_{0}}${\displaystyle g(z)=f(z)-w_{0}} will have the same number of roots (counted with multiplicity) in ${\displaystyle B}${\displaystyle B} as ${\displaystyle h(z):=f(z)-w_{1}}${\displaystyle h(z):=f(z)-w_{1}} for any ${\displaystyle w_{1}}${\displaystyle w_{1}} in ${\displaystyle D}${\displaystyle D}. This is because ${\displaystyle h(z)=g(z)+(w_{0}-w_{1})}${\displaystyle h(z)=g(z)+(w_{0}-w_{1})}, and for ${\displaystyle z}${\displaystyle z} on the boundary of ${\displaystyle B}${\displaystyle B}, ${\displaystyle |g(z)|\geq e>|w_{0}-w_{1}|}${\displaystyle |g(z)|\geq e>|w_{0}-w_{1}|}. Thus, for every ${\displaystyle w_{1}}${\displaystyle w_{1}} in ${\displaystyle D}${\displaystyle D}, there exists at least one ${\displaystyle z_{1}}${\displaystyle z_{1}} in ${\displaystyle B}${\displaystyle B} such that ${\displaystyle f(z_{1})=w_{1}}${\displaystyle f(z_{1})=w_{1}}. This means that the disk ${\displaystyle D}${\displaystyle D} is contained in ${\displaystyle f(B)}${\displaystyle f(B)}.

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

• Maximum modulus principle
• Rouché's theorem
• Schwarz lemma

• Open mapping theorem (functional analysis)

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

• Theorems in complex analysis

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• Articles containing proofs