SCHWEITZER 1985 Problem 9

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]