Koebe theorem

Koebe's covering theorem: There exist an absolute constant $ K > 0 $( the Koebe constant) such that if $ f \in S $( where $ S $ is the class of functions $ f ( z) = z + \dots $ that are regular and univalent in $ | z | < 1 $), then the set of values of the function $ w = f ( z) $ for $ | z | < 1 $ fills out the disc $ | w | < K $, where $ K $ is the largest number for which this holds. L. Bieberbach (1916) proved that $ K = 1 / 4 $ and that on the circle $ | w | = 1 / 4 $ there exists points not belonging to the image of the disc $ | z | < 1 $ under $ w = f ( z) $ only when

$$ f ( z) = \ \frac{z}{( 1 + e ^ {i \alpha } z ) ^ {2} } , $$

where $ \alpha $ is a real number. Koebe's covering theorem is sometimes stated as follows: If a function $ w = f ( z) $, $ f ( 0) = 0 $, is regular and univalent in $ | z | < 1 $ and maps the disc $ | z | < 1 $ onto a domain not containing the point $ c $, then $ | f ^ { \prime } ( 0) | \leq 4 c $.

Koebe's distortion theorems.

a) There exist positive numbers $ m _ {1} ( r) $, $ M _ {1} ( r) $, depending only on $ r $, such that for any $ f \in S $, $ | z | = r $,

$$ m _ {1} ( r) \leq | f ( z) | \leq M _ {1} ( r) . $$

b) There exists a number $ M ( r) $, depending only on $ r $, such that for $ f \in S $, $ | z _ {1} | , | z _ {2} | \leq r $,

$$ \frac{1}{M ( r) } \leq \ \left | \frac{f ^ { \prime } ( z _ {1} ) }{f ^ { \prime } ( z _ {2} ) } \right | \leq M ( r) . $$

This theorem can also be stated as follows: There exist positive numbers $ m _ {2} ( r) $, $ M _ {2} ( r) $, depending only on $ r $, such that for any $ f \in S $, $ | z | \leq r $,

$$ m _ {2} ( r) \leq | f ^ { \prime } ( z) | \leq M _ {2} ( r) . $$

Bieberbach proved that the best possible bounds in Koebe's distortion theorems are:

$$ m _ {1} ( r) = \ \frac{r}{( 1 + r ) ^ {2} } ,\ \ M _ {1} ( r) = \ \frac{r}{( 1 - r ) ^ {2} } , $$

$$ m _ {2} ( r) = \frac{1 - r }{( 1 + r ) ^ {3} } ,\ \ M _ {2} ( r) = \frac{1 + r }{( 1 - r ) ^ {3} } . $$

Koebe's theorems on mapping finitely-connected domains onto canonical domains.

a) Every $ n $- connected domain $ B $ of the $ z $- plane can be univalently mapped onto a circular domain (that is, onto a domain bounded by a finite number of complete non-intersecting circles; here some of the circles may degenerate to a point) of the $ \zeta $- plane. There exists just one normalized mapping among these mappings taking a given point $ z = a \in B $ to $ \zeta = \infty $ and such that the expansion of the mapping function in a neighbourhood of $ z = a $ has the form

$$ \frac{1}{z - a } + \alpha _ {1} ( z - a ) + \dots \ \textrm{ or } \ \ z + \frac{\alpha _ {1} }{z} + \dots , $$

according as $ a $ is finite or not.

b) Every $ n $- connected domain $ B $ of the $ z $- plane with boundary continua $ K _ {1} \dots K _ {n} $ can be univalently mapped onto the $ \zeta $- plane with $ n $ slits along arcs of logarithmic spirals with respective inclinations $ \theta _ {1} \dots \theta _ {n} $, $ 0 \leq \theta _ \nu \leq \pi / 2 $, $ \nu = 1 \dots n $, to the radial directions, and, moreover, such that the continuum $ K _ \nu $, $ \nu = 1 \dots n $, is taken to the arc with inclination $ \theta _ \nu $, the given points $ a , b \in B $ are taken to $ 0 $ and $ \infty $, and the expansion of the mapping function in a neighbourhood of $ z = b $ has the form

$$ \frac{1}{z - b } + \alpha _ {0} + \alpha _ {1} ( z - b ) + \dots \ \ \textrm{ or } \ \ z + \alpha _ {0} + \frac{\alpha _ {1} }{z} + \dots , $$

according as $ b $ is finite or not. The mapping is unique.

Theorems 1)–3) were established by P. Koebe (see –[4]).

References

[1a]	P. Koebe, "Ueber die Uniformisierung beliebiger analytischer Kurven" Nachr. K. Ges. Wissenschaft. Göttingen Math. Phys. Kl. , 2 (1907) pp. 191–210
[1b]	P. Koebe, "Ueber die Uniformisierung der algebraischen Kurven durch automorphe Funktionen mit imaginären Substitutionsgruppe" Nachr. K. Ges. Wissenschaft. Göttingen Math. Phys. Kl. , 4 (1908) pp. 68–76
[2]	P. Koebe, "Ueber die Uniformisierung der algebraischen Kurven II" Math. Ann. , 69 (1910) pp. 1–81
[3]	P. Koebe, "Abhandlung zur Theorie der konformen Abbildung IV" Acta Math. , 41 (1918) pp. 305–344
[4]	P. Koebe, "Abhandlung zur Theorie der konformen Abbildung V" Math. Z , 2 (1918) pp. 198–236
[5]	G.M. Goluzin, "Intrinsic problems in the theory of univalent functions" Uspekhi Mat. Nauk , 6 (1939) pp. 26–89 (In Russian)
[6]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[7]	J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958)

Comments

Koebe's covering theorem is related to Bloch's theorem: There exists an absolute constant $ B $ such that if $ f ( z) = z + a _ {2} z ^ {2} + \dots $ is analytic in $ D = \{ {z } : {| z | < 1 } \} $, then $ f ( D) $ contains a disc of radius $ B $ which is the one-to-one image of a subdomain of $ D $. The best (largest) value of $ B $ is called Bloch's constant. It is known that

$$ B \leq \frac{\Gamma ( 1/3) \Gamma ( 11/12) }{\sqrt {1 + \sqrt 3 } \Gamma ( 1/4) } , $$

and equality has been conjectured. For an up-to-date discussion of these matters, see [a1].

See also Landau theorems.

References

[a1]	C.D. Minda, "Bloch constants" J. d'Anal. Math. , 41 (1982) pp. 54–84
[a2]	J.B. Conway, "Functions of a complex variable" , Springer (1978)