Proof of Bieberbach's Conjecture

Year 1973, Volume: 22 , 0 - 0, 01.01.1973

C. Uluçay

https://doi.org/10.1501/Commua1_0000000626

Abstract

It is shown by the method of 2-dinıensional cross-section that for the class S of a- nalytic and schlicht functions
the inequaiity
/(z) = z + l-l< L
a.
İS always true, with equality for any n, n 2 if and only if/(s) is a Koebe func-
tion.
Survey. In this paper we prove the fam o us Bieberbach’s conjecture, i. e., for the class S of analytic and schlicht functions
f(z) ~ z a2Z^ + a,z^ + 1,
the inequality
i «n I n
I 2 I
is ahvays true, with equality for any n, n is a Koebe function
’ 2, if and only
(1—<>i't2)‘ = 2 + 2ei9 2^ -|- 3e2i6 z’ •••5 9 real-
Up to now, the conjecture has only been proved for re = 2, 3, 4 (see: [2], [3], [4], [5]).
As usual let V,n-1 be the set of pOints
a = ^^3, ®n-ı)
belonging to functions

Keywords

Bieberbach, Conjecture, Statistics

References

• Ankara Üniversitesi – Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Dergisi