Koebe theorem

Koebe's covering theorem: There exist an absolute constant (the Koebe constant) such that if (where is the class of functions that are regular and univalent in ), then the set of values of the function for fills out the disc , where is the largest number for which this holds. L. Bieberbach (1916) proved that and that on the circle there exists points not belonging to the image of the disc under only when

where is a real number. Koebe's covering theorem is sometimes stated as follows: If a function , , is regular and univalent in and maps the disc onto a domain not containing the point , then .

Koebe's distortion theorems.

a) There exist positive numbers , , depending only on , such that for any , ,

b) There exists a number , depending only on , such that for , ,

This theorem can also be stated as follows: There exist positive numbers , , depending only on , such that for any , ,

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

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

a) Every -connected domain of the -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 -plane. There exists just one normalized mapping among these mappings taking a given point to and such that the expansion of the mapping function in a neighbourhood of has the form

according as is finite or not.

b) Every -connected domain of the -plane with boundary continua can be univalently mapped onto the -plane with slits along arcs of logarithmic spirals with respective inclinations , , , to the radial directions, and, moreover, such that the continuum , , is taken to the arc with inclination , the given points are taken to and , and the expansion of the mapping function in a neighbourhood of has the form

according as 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 such that if is analytic in , then contains a disc of radius which is the one-to-one image of a subdomain of . The best (largest) value of is called Bloch's constant. It is known that

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)