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The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c0203001.png" /> of functions
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c0203002.png" /></td> </tr></table>
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that are regular in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c0203003.png" /> and have positive real part there. The class is named after C. Carathéodory, who determined the precise set of values of the system of coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c0203004.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c0203005.png" />, on the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c0203006.png" /> (see [[#References|[1]]], [[#References|[2]]]).
+
The class C $
 +
of functions
  
The Riesz–Herglotz theorem. In order that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c0203007.png" /> be of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c0203008.png" /> it is necessary and sufficient that it have a Stieltjes integral representation
+
$$
 +
f (z)  = 1 +
 +
\sum _ {n = 1 } ^  \infty 
 +
c _ {n} z  ^ {n}
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c0203009.png" /></td> </tr></table>
+
that are regular in the disc  $  | z | < 1 $
 +
and have positive real part there. The class is named after C. Carathéodory, who determined the precise set of values of the system of coefficients  $  \{ c _ {1} \dots c _ {n} \} $,
 +
$  n \geq  1 $,
 +
on the class  $  C $(
 +
see [[#References|[1]]], [[#References|[2]]]).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030010.png" /> is a non-decreasing function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030012.png" />.
+
The Riesz–Herglotz theorem. In order that  $  f (z) $
 +
be of class  $  C $
 +
it is necessary and sufficient that it have a Stieltjes integral representation
 +
 
 +
$$
 +
f (z)  = \
 +
\int\limits _ {- \pi } ^  \pi 
 +
 
 +
\frac{e  ^ {it} + z }{e  ^ {it} - z }
 +
\
 +
d \mu (t),
 +
$$
 +
 
 +
where $  \mu (t) $
 +
is a non-decreasing function on $  [- \pi , \pi ] $
 +
such that $  \mu ( \pi ) - \mu (- \pi ) = 1 $.
  
 
By means of this representation it is easy to deduce integral parametric representations for classes of functions which are convex and univalent in the disc, star-shaped and univalent in the disc, and others.
 
By means of this representation it is easy to deduce integral parametric representations for classes of functions which are convex and univalent in the disc, star-shaped and univalent in the disc, and others.
  
The Carathéodory–Toeplitz theorem. The set of values of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030014.png" />, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030015.png" /> is the closed convex bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030016.png" /> of points of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030017.png" />-dimensional complex Euclidean space at which the determinants
+
The Carathéodory–Toeplitz theorem. The set of values of the system $  \{ c _ {1} \dots c _ {n} \} $,  
 +
$  n \geq  1 $,  
 +
on $  C $
 +
is the closed convex bounded set $  K _ {n} $
 +
of points of the $  n $-
 +
dimensional complex Euclidean space at which the determinants
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030018.png" /></td> </tr></table>
+
$$
 +
\Delta _ {k}  = \
 +
\left |
  
are either all positive, or positive up to some number, beyond which they are all zero. In the latter case one obtains a face <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030019.png" /> of the coefficient body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030020.png" />. Corresponding to each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030021.png" /> there is just one function in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030022.png" />, which has the form
+
\begin{array}{llll}
 +
2  &c _ {1}  &\dots  &c _ {k}  \\
 +
{\overline{c}\; _ {1} }  & 2  &\dots  &c _ {k - 1 }  \\
 +
\cdot  &\cdot  &\dots  &\cdot  \\
 +
{\overline{c}\; _ {k} }  &{\overline{c}\; _ {k - 1 }  }  &\dots  & 2  \\
 +
\end{array}
 +
\
 +
\right | ,\ \
 +
1 \leq  k \leq  n,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030023.png" /></td> </tr></table>
+
are either all positive, or positive up to some number, beyond which they are all zero. In the latter case one obtains a face  $  \Pi _ {n} $
 +
of the coefficient body  $  K _ {n} $.
 +
Corresponding to each point of  $  \Pi _ {n} $
 +
there is just one function in the class $  C $,
 +
which has the form
 +
 
 +
$$
 +
f _ {N} (z)  = \
 +
\sum _ {j = 1 } ^ { N }
 +
\lambda _ {j}
 +
\frac{e ^ {it _ {j} } + z }{e ^ {it _ {j} } - z }
 +
,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030024.png" /></td> </tr></table>
+
$$
 +
\sum _ {j = 1 }  ^ {N}
 +
\lambda _ {j}  = 1,\ \
 +
\lambda _ {j}  > 0,\ \
 +
1 \leq  N \leq  n,\ \
 +
- \pi < t _ {j} \leq  \pi ,\ \
 +
t _ {j} \neq t _ {k}  $$
 +
 
 +
for  $  j \neq k $,
 +
$  k, j = 1 \dots N $.
 +
 
 +
The set of values of the coefficients  $  c _ {n} $,
 +
$  n = 1, 2 \dots $
 +
on  $  C $
 +
is the disc  $  | c _ {n} | \leq  2 $;  
 +
the only functions corresponding to the circle  $  | c _ {n} | = 2 $
 +
are
 +
 
 +
$$
 +
f (z)  = \
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030026.png" />.
+
\frac{e  ^ {it} + z }{e  ^ {it} - z }
 +
.
 +
$$
  
The set of values of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030028.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030029.png" /> is the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030030.png" />; the only functions corresponding to the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030031.png" /> are
+
The set of values of $  f (z _ {0} ) $(
 +
$  z _ {0} $
 +
fixed, $  | z _ {0} | < 1 $)
 +
on $  C $
 +
is the disc whose diameter is the interval  $  [ (1 - | z _ {0} | ) / (1 + | z _ {0} | ), (1 + | z _ {0} | ) / (1 - | z _ {0} | ) ] $;  
 +
the only functions corresponding to the boundary of this disc are
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030032.png" /></td> </tr></table>
+
$$
 +
f (z)  = \
  
The set of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030033.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030034.png" /> fixed, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030035.png" />) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030036.png" /> is the disc whose diameter is the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030037.png" />; the only functions corresponding to the boundary of this disc are
+
\frac{e  ^ {it} + z }{e  ^ {it} - z }
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030038.png" /></td> </tr></table>
+
Sets of values of systems of functionals of a more general type have also been considered (see [[#References|[6]]]). For the class  $  C $,
 +
variational formulas have been obtained by means of which a number of extremal problems in the class $  C $
 +
are solved by the functions  $  f _ {N} (z) $,
 +
$  N \geq  2 $(
 +
see [[#References|[6]]]).
  
Sets of values of systems of functionals of a more general type have also been considered (see [[#References|[6]]]). For the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030039.png" />, variational formulas have been obtained by means of which a number of extremal problems in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030040.png" /> are solved by the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030042.png" /> (see [[#References|[6]]]).
+
The main subclass of $  C $
 +
is the class  $  C _ {r} $
 +
of functions  $  f (z) \in C $
 +
having real coefficients  $  c _ {n} $,  
 +
$  n = 1, 2 , . . . $.  
 +
In order that  $  f (z) $
 +
belong to the class  $  C _ {r} $
 +
it is necessary and sufficient that it have a representation
  
The main subclass of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030043.png" /> is the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030044.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030045.png" /> having real coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030047.png" />. In order that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030048.png" /> belong to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030049.png" /> it is necessary and sufficient that it have a representation
+
$$
 +
f (z)  = \
 +
\int\limits _ { 0 } ^  \pi 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030050.png" /></td> </tr></table>
+
\frac{1 - z  ^ {2} }{1 - 2z  \cos  t + z  ^ {2} }
 +
\
 +
d \mu (t),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030051.png" /> is a non-decreasing function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030052.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030053.png" />. By means of this representation many extremal problems in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c02030054.png" /> are solved.
+
where $  \mu (t) $
 +
is a non-decreasing function on $  [0, \pi ] $
 +
such that $  \mu ( \pi ) - \mu (0) = 1 $.  
 +
By means of this representation many extremal problems in the class $  C _ {r} $
 +
are solved.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Carathéordory,  "Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen"  ''Math. Ann.'' , '''64'''  (1907)  pp. 95–115</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Carathéodory,  "Über den Variabilitätsbereich der Fourier'schen Konstanten von positiven harmonischen Funktionen"  ''Rend. Circ. Mat. Palermo'' , '''32'''  (1911)  pp. 193–217</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Toeplitz,  "Ueber die Fourier'sche Entwicklung positiver Funktionen"  ''Rend. Circ. Mat. Palermo'' , '''32'''  (1911)  pp. 191–192</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Riesz,  "Sur certains systèmes singuliers d'equations intégrales"  ''Ann. Sci. Ecole Norm. Super.'' , '''28'''  (1911)  pp. 33–62</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G. Herglotz,  "Über Potenzreihen mit positiven, reellen Teil im Einheitskreis"  ''Ber. Verhandl. Sächs. Akad. Wiss. Leipzig. Math.-Nat. Kl.'' , '''63'''  (1911)  pp. 501–511</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Carathéordory,  "Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen"  ''Math. Ann.'' , '''64'''  (1907)  pp. 95–115</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Carathéodory,  "Über den Variabilitätsbereich der Fourier'schen Konstanten von positiven harmonischen Funktionen"  ''Rend. Circ. Mat. Palermo'' , '''32'''  (1911)  pp. 193–217</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Toeplitz,  "Ueber die Fourier'sche Entwicklung positiver Funktionen"  ''Rend. Circ. Mat. Palermo'' , '''32'''  (1911)  pp. 191–192</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Riesz,  "Sur certains systèmes singuliers d'equations intégrales"  ''Ann. Sci. Ecole Norm. Super.'' , '''28'''  (1911)  pp. 33–62</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G. Herglotz,  "Über Potenzreihen mit positiven, reellen Teil im Einheitskreis"  ''Ber. Verhandl. Sächs. Akad. Wiss. Leipzig. Math.-Nat. Kl.'' , '''63'''  (1911)  pp. 501–511</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>

Latest revision as of 11:06, 30 May 2020


The class $ C $ of functions

$$ f (z) = 1 + \sum _ {n = 1 } ^ \infty c _ {n} z ^ {n} $$

that are regular in the disc $ | z | < 1 $ and have positive real part there. The class is named after C. Carathéodory, who determined the precise set of values of the system of coefficients $ \{ c _ {1} \dots c _ {n} \} $, $ n \geq 1 $, on the class $ C $( see [1], [2]).

The Riesz–Herglotz theorem. In order that $ f (z) $ be of class $ C $ it is necessary and sufficient that it have a Stieltjes integral representation

$$ f (z) = \ \int\limits _ {- \pi } ^ \pi \frac{e ^ {it} + z }{e ^ {it} - z } \ d \mu (t), $$

where $ \mu (t) $ is a non-decreasing function on $ [- \pi , \pi ] $ such that $ \mu ( \pi ) - \mu (- \pi ) = 1 $.

By means of this representation it is easy to deduce integral parametric representations for classes of functions which are convex and univalent in the disc, star-shaped and univalent in the disc, and others.

The Carathéodory–Toeplitz theorem. The set of values of the system $ \{ c _ {1} \dots c _ {n} \} $, $ n \geq 1 $, on $ C $ is the closed convex bounded set $ K _ {n} $ of points of the $ n $- dimensional complex Euclidean space at which the determinants

$$ \Delta _ {k} = \ \left | \begin{array}{llll} 2 &c _ {1} &\dots &c _ {k} \\ {\overline{c}\; _ {1} } & 2 &\dots &c _ {k - 1 } \\ \cdot &\cdot &\dots &\cdot \\ {\overline{c}\; _ {k} } &{\overline{c}\; _ {k - 1 } } &\dots & 2 \\ \end{array} \ \right | ,\ \ 1 \leq k \leq n, $$

are either all positive, or positive up to some number, beyond which they are all zero. In the latter case one obtains a face $ \Pi _ {n} $ of the coefficient body $ K _ {n} $. Corresponding to each point of $ \Pi _ {n} $ there is just one function in the class $ C $, which has the form

$$ f _ {N} (z) = \ \sum _ {j = 1 } ^ { N } \lambda _ {j} \frac{e ^ {it _ {j} } + z }{e ^ {it _ {j} } - z } , $$

where

$$ \sum _ {j = 1 } ^ {N} \lambda _ {j} = 1,\ \ \lambda _ {j} > 0,\ \ 1 \leq N \leq n,\ \ - \pi < t _ {j} \leq \pi ,\ \ t _ {j} \neq t _ {k} $$

for $ j \neq k $, $ k, j = 1 \dots N $.

The set of values of the coefficients $ c _ {n} $, $ n = 1, 2 \dots $ on $ C $ is the disc $ | c _ {n} | \leq 2 $; the only functions corresponding to the circle $ | c _ {n} | = 2 $ are

$$ f (z) = \ \frac{e ^ {it} + z }{e ^ {it} - z } . $$

The set of values of $ f (z _ {0} ) $( $ z _ {0} $ fixed, $ | z _ {0} | < 1 $) on $ C $ is the disc whose diameter is the interval $ [ (1 - | z _ {0} | ) / (1 + | z _ {0} | ), (1 + | z _ {0} | ) / (1 - | z _ {0} | ) ] $; the only functions corresponding to the boundary of this disc are

$$ f (z) = \ \frac{e ^ {it} + z }{e ^ {it} - z } . $$

Sets of values of systems of functionals of a more general type have also been considered (see [6]). For the class $ C $, variational formulas have been obtained by means of which a number of extremal problems in the class $ C $ are solved by the functions $ f _ {N} (z) $, $ N \geq 2 $( see [6]).

The main subclass of $ C $ is the class $ C _ {r} $ of functions $ f (z) \in C $ having real coefficients $ c _ {n} $, $ n = 1, 2 , . . . $. In order that $ f (z) $ belong to the class $ C _ {r} $ it is necessary and sufficient that it have a representation

$$ f (z) = \ \int\limits _ { 0 } ^ \pi \frac{1 - z ^ {2} }{1 - 2z \cos t + z ^ {2} } \ d \mu (t), $$

where $ \mu (t) $ is a non-decreasing function on $ [0, \pi ] $ such that $ \mu ( \pi ) - \mu (0) = 1 $. By means of this representation many extremal problems in the class $ C _ {r} $ are solved.

References

[1] C. Carathéordory, "Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen" Math. Ann. , 64 (1907) pp. 95–115
[2] C. Carathéodory, "Über den Variabilitätsbereich der Fourier'schen Konstanten von positiven harmonischen Funktionen" Rend. Circ. Mat. Palermo , 32 (1911) pp. 193–217
[3] O. Toeplitz, "Ueber die Fourier'sche Entwicklung positiver Funktionen" Rend. Circ. Mat. Palermo , 32 (1911) pp. 191–192
[4] F. Riesz, "Sur certains systèmes singuliers d'equations intégrales" Ann. Sci. Ecole Norm. Super. , 28 (1911) pp. 33–62
[5] G. Herglotz, "Über Potenzreihen mit positiven, reellen Teil im Einheitskreis" Ber. Verhandl. Sächs. Akad. Wiss. Leipzig. Math.-Nat. Kl. , 63 (1911) pp. 501–511
[6] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
How to Cite This Entry:
Carathéodory class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carath%C3%A9odory_class&oldid=22237
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article