mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Carathéodory kernel theorem is a result in

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

and

geometric function theory Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem. Topics in geometric function theory The following are some of the most important topics in ge ...

established by the Greek mathematician

Constantin Carathéodory Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...

in 1912. The

uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...

on compact sets of a sequence of

holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...

univalent functions In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective. Examples The function f \colon z \mapsto 2z + z^2 is univalent in the open unit disc, ...

, defined on the

unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose di ...

in the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

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 In mathematics, the Loewner differential equation, or Loewner equation, is an ordinary differential equation discovered by Charles Loewner in 1923 in complex analysis and geometric function theory. Originally introduced for studying slit mappings (c ...

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 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

\

. Thus the kernel is either a connected open set containing 0 or the one point set

\

. 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 ''U''₁ ∩ ''U''₂ ∩ ..., the sequence converges to the component of the interior containing 0. Otherwise, if 0 is not an interior point, the sequence converges to

\

Kernel theorem

Let ''f''_''n''(''z'') be a sequence of

univalent function In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective. Examples The function f \colon z \mapsto 2z + z^2 is univalent in the open unit disc, ...

s 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

\

, 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 In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after French mathematician Paul Montel, and give conditions under which a family of holomorphic fu ...

, 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 In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following: Koebe Quarter Theorem. The image of an injective analytic function f:\mathbf\to\mathbb from the unit disk \mathbf onto a subset of the complex plane contains ...

''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 In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following: Koebe Quarter Theorem. The image of an injective analytic function f:\mathbf\to\mathbb from the unit disk \mathbf onto a subset of the complex plane contains ...

, f_n(z),  \le f_n^\prime(0) .

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 In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected space, simply connected open set, open subset of the complex plane, complex number plane C which is not all of C, then there exists a biholomorphy ...

forces ''f'' = ''g'', so the original sequence ''f''_''n'' is uniformly convergent on compact sets.

References

* * * {{DEFAULTSORT:Caratheodory kernel theorem Theorems in complex analysis