Carathéodory's theorem (conformal mapping)

Carathéodory's theorem (conformal mapping) In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, first proved in 1913,[citation needed] states that the conformal mapping sending the unit disk to the region in the complex plane bounded by a Jordan curve extends continuously to a homeomorphism from the unit circle onto the Jordan curve. The result is one of Carathéodory's results on prime ends and the boundary behaviour of univalent holomorphic functions.

Contents 1 Proofs of Carathéodory's theorem 2 Continuous extension and the Carathéodory-Torhorst theorem 3 Notes 4 References Proofs of Carathéodory's theorem The first proof of Carathéodory's theorem presented here is a summary of the short self-contained account in Garnett & Marshall (2005, pp. 14–15); there are related proofs in Pommerenke (1992) and Krantz (2006).

Carathéodory's theorem. If f maps the open unit disk D conformally onto a bounded domain U in C, then f has a continuous one-to-one extension to the closed unit disk if and only if ∂U is a Jordan curve.

Clearly if f admits an extension to a homeomorphism, then ∂U must be a Jordan curve.

Conversely if ∂U is a Jordan curve, the first step is to prove f extends continuously to the closure of D. In fact this will hold if and only if f is uniformly continuous on D: for this is true if it has a continuous extension to the closure of D; and, if f is uniformly continuous, it is easy to check f has limits on the unit circle and the same inequalities for uniform continuity hold on the closure of D.

Suppose that f is not uniformly continuous. In this case there must be an ε > 0 and a point ζ on the unit circle and sequences zn, wn tending to ζ with |f(zn) − f(wn)| ≥ 2ε. This is shown below to lead to a contradiction, so that f must be uniformly continuous and hence has a continuous extension to the closure of D.

For 0 < r < 1, let γr be the curve given by the arc of the circle | z − ζ | = r lying within D. Then f ∘ γr is a Jordan curve. Its length can be estimated using the Cauchy–Schwarz inequality: {displaystyle displaystyle {ell (fcirc gamma _{r})=int _{gamma _{r}}|f^{prime }(z)|,|dz|leq left(int _{gamma _{r}},|dz|right)^{1/2}cdot left(int _{gamma _{r}}|f^{prime }(z)|^{2},|dz|right)^{1/2}leq (2pi r)^{1/2}cdot left(int _{{theta :|zeta +re^{itheta }|<1}}|f^{prime }(zeta +re^{itheta })|^{2},r,dtheta right)^{1/2}.}} Hence there is a "length-area estimate": {displaystyle displaystyle {int _{0}^{1}ell (fcirc gamma _{r})^{2},{dr over r}leq 2pi int _{|z|<1}|f^{prime }(z)|^{2},dx,dy=2pi cdot {rm {Area}},f(D)

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