The theorem, published by Carathéodory in 1913, states that any conformal mapping sending the unit disk to some 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.
The first proof of Carathéodory's theorem presented here is a summary of the short self-contained account in Garnett & Marshall (2005, pp.
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.
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: Hence there is a "length-area estimate": The finiteness of the integral on the left hand side implies that there is a sequence rn decreasing to 0 with
therefore implies that the curve has limiting points an, bn at its two ends with
These two limit points must lie on ∂U, because f is a homeomorphism between D and U and thus a sequence converging in U has to be the image under f of a sequence converging in D. By assumption there exist a homeomorphism β between the circle ∂D and ∂U.
Since β−1 is uniformly continuous, the distance between the two points ξn and ηn corresponding to an and bn in ∂U must tend to 0.
So eventually the smallest circular arc in ∂D joining ξn and ηn is defined.
Denote τn image of this arc under β.
By uniform continuity of β, diameter of τn in ∂U tends to 0.
Together τn and f ∘ γrn form a simple Jordan curve.
Its interior Un is contained in U by the Jordan curve theorem for ∂U and ∂Un: to see this, notice that U is the interior of ∂U, as it is bounded, connected and it is both open and closed in the complement of ∂U; so the exterior region of ∂U is unbounded, connected and does not intersect ∂Un, hence its closure is contained in the closure of the exterior of ∂Un; taking complements, we get the desired inclusion.
Indeed, the arc γrn divides D into Vn and complementary region
, so under the conformal homeomorphism f the curve f ∘ γrn divides U into
On the other hand, passing to subsequences of (zn) and (wn) if necessary, it may be assumed that zn and wn both lie in Vn.
Arguing as before, its interior V is contained in U and is a connected component of U \ f(X ∪ Y).
On the other hand, D \ (X ∪ Y) is the disjoint union of two open sectors W1 and W2.
Hence, for one of them, W1 say, f(W1) = V. Let Z be the portion of ∂W1 on the unit circle, so that Z is a closed arc and f(Z) is a subset of both ∂U and the closure of V. But their intersection is a single point and hence f is constant on Z.
Since non-constant holomorphic functions have isolated zeros, this forces f to be constant, a contradiction.
The first proof follows Carathéodory's original method of proof from 1913 using properties of Lebesgue measure on the circle: the continuous extension of the inverse function g of f to ∂U is justified by Fatou's theorem on the boundary behaviour of bounded harmonic functions on the unit disk.
The second proof is based on the method of Lindelöf (1914), where a sharpening of the maximum modulus inequality was established for bounded holomorphic functions h defined on a bounded domain V: if a lies in V, then where 0 ≤ t ≤ 1, M is maximum modulus of h for sequential limits on ∂U and m is the maximum modulus of h for sequential limits on ∂U lying in a sector centred on a subtending an angle 2πt at a.
[3] An extension of the theorem states that a conformal isomorphism where
is a simply connected subset of the Riemann sphere, extends continuously to the unit circle if and only if the boundary of
This result is often also attributed to Carathéodory, but was first stated and proved by Marie Torhorst in her 1918 thesis,[4] under the supervision of Hans Hahn, using Carathéodory's theory of prime ends.
More precisely, Torhorst proved that local connectivity is equivalent to the domain having only prime ends of the first kind.
By the theory of prime ends, the latter property, in turn, is equivalent to