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''on conformal mapping of domains with variable boundaries''
 
''on conformal mapping of domains with variable boundaries''
  
 
One of the main results in the theory of conformal mapping of domains with variable boundaries; obtained by C. Carathéodory [[#References|[1]]].
 
One of the main results in the theory of conformal mapping of domains with variable boundaries; obtained by C. Carathéodory [[#References|[1]]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c0203401.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c0203402.png" /> be a sequence of simply-connected domains of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c0203403.png" />-plane containing a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c0203404.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c0203405.png" />. If there exists a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c0203406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c0203407.png" />, belonging to all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c0203408.png" />, then the kernel of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c0203409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034010.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034011.png" /> is the largest domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034012.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034013.png" /> and such that for each compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034014.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034015.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034017.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034019.png" />. A largest domain is one which contains any other domain having the same property. If there is no such a disc, then by the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034020.png" /> of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034022.png" /> one means the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034023.png" /> (in this case one says that the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034025.png" /> has a degenerate kernel). A sequence of domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034027.png" /> converges to a kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034028.png" /> if any subsequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034029.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034030.png" /> as its kernel.
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Let $B_n$, $n=1,2,\dots,$ be a sequence of simply-connected domains of the $z$-plane containing a fixed point $z_0$, $z_0\neq\infty$. If there exists a disc $|z-z_0|<\rho$, $\rho>0$, belonging to all $B_n$, then the kernel of the sequence $B_n$, $n=1,2,\dots,$ with respect to $z_0$ is the largest domain $B$ containing $z_0$ and such that for each compact set $E$ belonging to $B$ there is an $N$ such that $E$ belongs to $B_n$ for all $n\geq N$. A largest domain is one which contains any other domain having the same property. If there is no such a disc, then by the kernel $B$ of the sequence $B_n$, $n=1,2,\dots,$ one means the point $z_0$ (in this case one says that the sequence $B_n$, $n=1,2,\dots,$ has a degenerate kernel). A sequence of domains $B_n$, $n=1,2,\dots,$ converges to a kernel $B$ if any subsequence of $B_n$ has $B$ as its kernel.
  
Carathéodory's theorem. Suppose that one is given a sequence of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034034.png" /> that are regular and univalent in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034035.png" /> and that map this disc onto the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034036.png" />, respectively. Then in order that the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034038.png" /> converges in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034039.png" /> to a finite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034040.png" />, it is necessary and sufficient that the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034042.png" /> converges to a kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034043.png" /> which is either the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034044.png" /> or a domain containing more than one boundary point. Moreover, the convergence is uniform on compact sets in the interior of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034045.png" />. If the limit function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034046.png" />, then it maps the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034047.png" /> univalently onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034048.png" />, and the inverse functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034050.png" /> are uniformly convergent on compact sets in the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034051.png" /> to the inverse function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034052.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034053.png" />.
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Carathéodory's theorem. Suppose that one is given a sequence of functions $z=f_n(\zeta)$, $f_n(\zeta_0)=z_0$, $f'_n(\zeta_0)>0$, $n=1,2,\dots,$ that are regular and univalent in the disc $|\zeta-\zeta_0|<1$ and that map this disc onto the domains $B_n$, respectively. Then in order that the sequence $f_n(\zeta)$, $n=1,2,\dots,$ converges in the disc $|\zeta-\zeta_0|<1$ to a finite function $f(\zeta)$, it is necessary and sufficient that the sequence $B_n$, $n=1,2,\dots,$ converges to a kernel $B$ which is either the point $z_0$ or a domain containing more than one boundary point. Moreover, the convergence is uniform on compact sets in the interior of the disc $|\zeta-\zeta_0|<1$. If the limit function $f(\zeta)\not\equiv\mathrm{const}$, then it maps the disc $|\zeta-\zeta_0|<1$ univalently onto $B$, and the inverse functions $\phi_n(z)$, $n=1,2,\dots,$ are uniformly convergent on compact sets in the interior of $B$ to the inverse function $\phi(z)$ of $f(\zeta)$.
  
The question of the convergence of univalent functions in multiply-connected domains is considered analogously. One such theorem is given below for unbounded domains. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034055.png" /> be a sequence of domains in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034056.png" />-plane containing some fixed neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034057.png" />. Then the kernel of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034059.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034060.png" /> is the largest domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034061.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034062.png" /> and such that any closed subdomain of it is a subset of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034063.png" /> from some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034064.png" /> onwards. Convergence of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034066.png" /> to the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034067.png" /> is defined as before. The following theorem holds [[#References|[2]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034069.png" /> be a sequence of domains in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034070.png" />-plane containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034071.png" /> and converging to a kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034072.png" />, and suppose that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034074.png" /> map them univalently onto corresponding domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034075.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034076.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034079.png" />. Then in order that the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034081.png" /> converges uniformly on compact sets in the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034082.png" /> to a univalent function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034083.png" />, it is necessary and sufficient that the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034085.png" /> possesses a kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034086.png" /> and converges to it. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034087.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034088.png" /> univalently onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020340/c02034089.png" />.
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The question of the convergence of univalent functions in multiply-connected domains is considered analogously. One such theorem is given below for unbounded domains. Let $B_n$, $n=1,2,\dots,$ be a sequence of domains in the $z$-plane containing some fixed neighbourhood of $z=\infty$. Then the kernel of the sequence $B_n$, $n=1,2,\dots,$ with respect to $z=\infty$ is the largest domain $B$ containing $z=\infty$ and such that any closed subdomain of it is a subset of all $B_n$ from some $n$ onwards. Convergence of the sequence $B_n$, $n=1,2,\dots,$ to the kernel $B$ is defined as before. The following theorem holds [[#References|[2]]]. Let $A_n$, $n=1,2,\dots,$ be a sequence of domains in the $z$-plane containing $z=\infty$ and converging to a kernel $A$, and suppose that the functions $\zeta=f_n(z)$, $n=1,2,\dots,$ map them univalently onto corresponding domains $B_n$ containing $\zeta=\infty$; $f_n(\infty)=\infty$, $f'_n(\infty)=1$, $n=1,2,\dots$. Then in order that the sequence $f_n(z)$, $n=1,2,\dots,$ converges uniformly on compact sets in the interior of $A$ to a univalent function $f(z)$, it is necessary and sufficient that the sequence $B_n$, $n=1,2,\dots,$ possesses a kernel $B$ and converges to it. In this case $\zeta=f(z)$ maps $A$ univalently onto $B$.
  
 
It is possible to give other theorems on the convergence of a sequence of univalent functions, in dependence of the method of normalizing them (see [[#References|[2]]]).
 
It is possible to give other theorems on the convergence of a sequence of univalent functions, in dependence of the method of normalizing them (see [[#References|[2]]]).

Latest revision as of 21:37, 22 December 2018

on conformal mapping of domains with variable boundaries

One of the main results in the theory of conformal mapping of domains with variable boundaries; obtained by C. Carathéodory [1].

Let $B_n$, $n=1,2,\dots,$ be a sequence of simply-connected domains of the $z$-plane containing a fixed point $z_0$, $z_0\neq\infty$. If there exists a disc $|z-z_0|<\rho$, $\rho>0$, belonging to all $B_n$, then the kernel of the sequence $B_n$, $n=1,2,\dots,$ with respect to $z_0$ is the largest domain $B$ containing $z_0$ and such that for each compact set $E$ belonging to $B$ there is an $N$ such that $E$ belongs to $B_n$ for all $n\geq N$. A largest domain is one which contains any other domain having the same property. If there is no such a disc, then by the kernel $B$ of the sequence $B_n$, $n=1,2,\dots,$ one means the point $z_0$ (in this case one says that the sequence $B_n$, $n=1,2,\dots,$ has a degenerate kernel). A sequence of domains $B_n$, $n=1,2,\dots,$ converges to a kernel $B$ if any subsequence of $B_n$ has $B$ as its kernel.

Carathéodory's theorem. Suppose that one is given a sequence of functions $z=f_n(\zeta)$, $f_n(\zeta_0)=z_0$, $f'_n(\zeta_0)>0$, $n=1,2,\dots,$ that are regular and univalent in the disc $|\zeta-\zeta_0|<1$ and that map this disc onto the domains $B_n$, respectively. Then in order that the sequence $f_n(\zeta)$, $n=1,2,\dots,$ converges in the disc $|\zeta-\zeta_0|<1$ to a finite function $f(\zeta)$, it is necessary and sufficient that the sequence $B_n$, $n=1,2,\dots,$ converges to a kernel $B$ which is either the point $z_0$ or a domain containing more than one boundary point. Moreover, the convergence is uniform on compact sets in the interior of the disc $|\zeta-\zeta_0|<1$. If the limit function $f(\zeta)\not\equiv\mathrm{const}$, then it maps the disc $|\zeta-\zeta_0|<1$ univalently onto $B$, and the inverse functions $\phi_n(z)$, $n=1,2,\dots,$ are uniformly convergent on compact sets in the interior of $B$ to the inverse function $\phi(z)$ of $f(\zeta)$.

The question of the convergence of univalent functions in multiply-connected domains is considered analogously. One such theorem is given below for unbounded domains. Let $B_n$, $n=1,2,\dots,$ be a sequence of domains in the $z$-plane containing some fixed neighbourhood of $z=\infty$. Then the kernel of the sequence $B_n$, $n=1,2,\dots,$ with respect to $z=\infty$ is the largest domain $B$ containing $z=\infty$ and such that any closed subdomain of it is a subset of all $B_n$ from some $n$ onwards. Convergence of the sequence $B_n$, $n=1,2,\dots,$ to the kernel $B$ is defined as before. The following theorem holds [2]. Let $A_n$, $n=1,2,\dots,$ be a sequence of domains in the $z$-plane containing $z=\infty$ and converging to a kernel $A$, and suppose that the functions $\zeta=f_n(z)$, $n=1,2,\dots,$ map them univalently onto corresponding domains $B_n$ containing $\zeta=\infty$; $f_n(\infty)=\infty$, $f'_n(\infty)=1$, $n=1,2,\dots$. Then in order that the sequence $f_n(z)$, $n=1,2,\dots,$ converges uniformly on compact sets in the interior of $A$ to a univalent function $f(z)$, it is necessary and sufficient that the sequence $B_n$, $n=1,2,\dots,$ possesses a kernel $B$ and converges to it. In this case $\zeta=f(z)$ maps $A$ univalently onto $B$.

It is possible to give other theorems on the convergence of a sequence of univalent functions, in dependence of the method of normalizing them (see [2]).

References

[1] C. Carathéodory, "Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten" Math. Ann. , 72 (1912) pp. 107–144
[2] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)


Comments

References

[a1] P.L. Duren, "Univalent functions" , Springer (1983) pp. Chapt. 3
How to Cite This Entry:
Carathéodory theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carath%C3%A9odory_theorem&oldid=22251
This article was adapted from an original article by G.V. Kuz'mina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article