Carathéodory class

The class of functions

that are regular in the disc 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 , , on the class (see [1], [2]).

The Riesz–Herglotz theorem. In order that be of class it is necessary and sufficient that it have a Stieltjes integral representation

where is a non-decreasing function on such that .

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 , , on is the closed convex bounded set of points of the -dimensional complex Euclidean space at which the determinants

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

where

for , .

The set of values of the coefficients , on is the disc ; the only functions corresponding to the circle are

The set of values of ( fixed, ) on is the disc whose diameter is the interval ; the only functions corresponding to the boundary of this disc are

Sets of values of systems of functionals of a more general type have also been considered (see [6]). For the class , variational formulas have been obtained by means of which a number of extremal problems in the class are solved by the functions , (see [6]).

The main subclass of is the class of functions having real coefficients , . In order that belong to the class it is necessary and sufficient that it have a representation

where is a non-decreasing function on such that . By means of this representation many extremal problems in the class 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)