Analytic maps from upper half plane to itself

Question

I know that the bilinear map $f(z)=\frac{az+b}{cz+d}, a,b,c,d\in \mathbb{R}, ad-bc>0$, maps the upper half plane onto itself. But is every analytic function that does so is of this form?

Answer 1

Regardless of the correctness of the sign in the question, the answer is no, meaning that there are analytic functions from the upper half plane onto itself which are not Möbius transformations. See for instance this answer or the book The Problem of Moments by Shohat and Tamarkin.