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)