Carathéodory kernel theorem

In mathematics, the Carathéodory kernel theorem is a result in complex analysis and geometric function theory established by the Greek mathematician Constantin Carathéodory in 1912. The uniform convergence on compact sets of a sequence of holomorphic univalent functions, defined on the unit disk in the complex plane and fixing 0, can be formulated purely geometrically in terms of the limiting behaviour of the images of the functions. The kernel theorem has wide application in the theory of univalent functions and in particular provides the geometric basis for the Loewner differential equation.

Kernel of a sequence of open sets

Let U_n be a sequence of open sets in C containing 0. Let V_n be the connected component of the interior of U_n ∩ U_{n + 1} ∩ ... containing 0. The kernel of the sequence is defined to be the union of the V_n's, provided it is non-empty; otherwise it is defined to be [math]\displaystyle{ \{0\} }[/math]. Thus the kernel is either a connected open set containing 0 or the one point set [math]\displaystyle{ \{0\} }[/math]. The sequence is said to converge to a kernel if each subsequence has the same kernel.

Examples

• If U_n is an increasing sequence of connected open sets containing 0, then the kernel is just the union.
• If U_n is a decreasing sequence of connected open sets containing 0, then, if 0 is an interior point of U₁ ∩ U₂ ∩ ..., the sequence converges to the component of the interior containing 0. Otherwise, if 0 is not an interior point, the sequence converges to [math]\displaystyle{ \{0\} }[/math].

Kernel theorem

Let f_n(z) be a sequence of holomorphic univalent functions on the unit disk D, normalised so that f_n(0) = 0 and f '_n (0) > 0. Then f_n converges uniformly on compacts in D to a function f if and only if U_n = f_n(D) converges to its kernel and this kernel is not C. If the kernel is [math]\displaystyle{ \{0\} }[/math], then f = 0. Otherwise the kernel is a connected open set U, f is univalent on D and f(D) = U.

Proof

Using Hurwitz's theorem and Montel's theorem, it is straightforward to check that if f_n tends uniformly on compacta to f then each subsequence of U_n has kernel U = f(D).

Conversely if U_n converges to a kernel not equal to C, then by the Koebe quarter theorem U_n contains the disk of radius f '_n(0) / 4 with centre 0. The assumption that U ≠ C implies that these radii are uniformly bounded. By the Koebe distortion theorem

[math]\displaystyle{ |f_n(z)| \le f_n^\prime(0) {|z|\over (1-|z|)^2}. }[/math]

Hence the sequence f_n is uniformly bounded on compact sets. If two subsequences converge to holomorphic limits f and g, then f(0) = g(0) and with f'(0), g'(0) ≥ 0. By the first part and the assumptions it follows that f(D) = g(D). Uniqueness in the Riemann mapping theorem forces f = g, so the original sequence f_n is uniformly convergent on compact sets.

References

• Carathéodory, C. (1912), "Untersuchungen über die konformen Abbildungen von festen und veranderlichen Gebieten", Math. Ann. 72: 107–144, doi:10.1007/bf01456892, https://zenodo.org/record/1428282/files/article.pdf 
• Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, ISBN 0-387-90795-5 
• Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, 15, Vandenhoeck & Ruprecht