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Bieberbach の予想

We can find the proof in
Chapter 9 with prepareations in Chapter 7 and 8
in the book:
Topics in Hardy Classes and Univalent Functions,
Birkhauser Advanced Texts,
Marvin Rosenblum and James Rovnyak

Louis de Branges solved this conjecture in 1984.
Actually, he has proved the Milin conjecture.
If someone had solved Millin conjecture,
then he would have solved the Bieberbach's one itself.
In 1916, Bieberbach proposed "Bieberbach conjecture".
In 1923,Lowner proved case n=2 by the parametric method.
In 1955 Garabedian and Schiffer proved the case n=4.
In 1960 Charzynski and Schiffer proved the case n=4 by using Grunsky's inequality without the method Garanbedian and his colleagues used.
In 1968 Pederson and Ozawa independently proved the case n=6.
In 1972 Pederson and Schiffer proved the case n=5 by usin new generalized Grunsky's inequality.
Unfortuntely some mistake was found but was corrected later.
And finally, these tries came up against difficulties.
In 1936 Robertson proposed his conjecture.
In 1971 Millin in Russia proposed his conjecture which includes the Robertson conjecture.


Cardinarity(less than or equal to)

The cardinarity of set A is said to be less than or equal to the cardinarity of set B if there is an injective transformation of set A into set B.

Cardinarity( of a finite set)

de Branges は今生きている数学者としては最高の人である。評価は分かれるが、

Cardinarity is a measure of size for sets.

The cardinarity of a finite set is a nonnegative integer n which compares the set with the set of nonnegative integers less than n.

これは一見無用な定義に見えるが、cardinarity は、何かとなにかを比較する相対的なものにより定義されることを言っている。つまり無限集合のサイズを言うときに有限集合を例外としてではなく、決まりきった方法を土台にしたいためにこのような言い回しにしている。