# Carathéodory class

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=46205
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article