I am trying to use the fact that every bi- holomorphic function from the unit disk to itself can be writen as $$ \varphi(z)=\frac{\lambda(z-a)}{1-\bar{a} z} $$ for every $a,\lambda \in D_1(0)$ to show that mobius transformation from the upper half plane $\{z:\mathrm{Im}(z) > 0\}$ to iteself can be written as $$f(z) = \frac{az+b}{cz+d}$$ s.t $a,b,c,d \in \mathbb{R}$
How do I use $\varphi(z)$ for that? not sure how do I solve it without 3 system of equations