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Miklós Schweitzer 1985, Problem 9

Source: Miklós Schweitzer 1985

September 5, 2016

Miklos Schweitzercollege contestscomplex analysisfunction

Problem Statement

Let

D=\{ z\in \mathbb C\colon |z|<1\}

and

D=\{ w\in \mathbb C \colon |w|=1\}

. Prove that if for a function

f\colon D\times B\rightarrow\mathbb C

the equality

f\left( \frac{az+b}{\overline{b}z+\overline{a}}, \frac{aw+b}{\overline{b}w+\overline a} \right)=f(z,w)+f\left(\frac{b}{\overline a}, \frac{aw+b}{\overline b w+\overline a} \right)

holds for all

z\in D, w\in B

and

a, b\in \mathbb C,|a|^2=|b|^2+1

, then there is a function

L\colon (0, \infty )\rightarrow \mathbb C

satisfying

L(pq)=L(p)+L(q)\,\,\,\text{for all}\,\,\, p,q > 0

such that

f

can be represented as

f(z,w)=L\left( \frac{1-|z|^2}{|w-z|^2}\right)\,\,\,\text{for all}\,\,\, z\in D, w\in B

. [Gy. Maksa]

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