View source for Carathéodory class

The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020300/c0203001.png" /> of functions

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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 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

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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" />.

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

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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

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where

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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" />.

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

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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

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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 <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

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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.

====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>