The question asks:
Let $\varphi$ be a linear fractional transformation which maps the upper half plane $\{z : \operatorname{Im}(z) > 0\}$ onto itself. Prove that if there exist distinct $z_1$ and $z_2$ having positive imaginary parts with $\varphi(z_1) = z_1$ and $\varphi(z2) = z_2$ then $\varphi(z) = z$ for all $z$.
There is an apparently related question that asks: 2.) Let $\varphi$ be a linear fractional transformation which maps the unit disk $\{z : |z| < 1\}$ onto itself. Prove that if there exist distinct $z_1$ and $z_2$ in the disk with $\varphi(z_1)=z_1$ and $\varphi(z_2)=z_2$ then $\varphi(z)=z$ for all $z$.
I was wondering if there is some sort of easy approach to this type of question that I'm missing? I don't believe they were intended to be particularly difficult, but after a significant amount of time I've run out of good ideas.