It is well-known that the biholomorphic self-maps of the upper half-plane are Mobius transformations $$\dfrac{az+b}{cz+d}$$ with $a, b, c, d\in\mathbb{R}$ and $ad-bc=1.$
Also, on the unit disk, holomorphic self-maps that extend continuously to the boundary are the finite Blaschke products. These are a finite product of disk automorphisms up to a rotation.
I would like to see some kind of nice representation theorem for holomorphic self-maps of the upper half-plane. I tried the composition $\varphi B\varphi^{-1},$ where $\varphi$ is a Mobius transformation from the unit disk to the upper half-plane and $B$ is a finite Blaschke product. Even for degree two Blaschke product $B,$ this gives a complicated-looking formula from which I cannot see anything.
- Is there a nice way to represent holomorphic self-maps of the upper half-plane? I would like to see this with a proof.
Added later: I conjecture that one possible representation is $$f(z)+\sum_{1\le j\le n}\dfrac{c_j}{z-a_j}$$ where $a_j\in\mathbb{R}, c_j\in\mathbb{R}^-$ and $f$ is some holomorphic self-map on the closed upper half plane. For example, $f(z)=az+b$ with $a\in\mathbb{R}^+,b\in\mathbb{R}$ works. See the comment section for a vague explanation of how I arrived at this. What is the most general form of $f,$ and how can we prove this in general?
Best Answer
The analytic functions mapping the upper half plane $H$ into itself are called the Pick functions and they have the form
$$f(z)=kz+\alpha-\int_{-\infty}^{\infty}\frac{z\gamma+1}{z-\gamma}\mu(d\gamma)$$ where $k\geq 0$ and $\mu$ is a positive bounded mesure. Furthermore, there are such that $\lim_{y\to 0} \Im f(x+iy)=0$ -thus preserving the real axis -if and only if $\mu$ is purely singular- this includes atomic measures which are what you need and also Cantor like ones. If $\mu$ is a finite sum of Dirac, you get the analogs of Blaschke products.
I had to use these Pick functions for writing 'Which functions preserve Cauchy laws' Proc. Amer. Math. Soc, 1977, Vol 67 pages 277-286. I was unaware of Pick functions and a bit painfully I had, like you, to move from the disk to $H$ where the necessary theory for the disk is developped in the Rudin's book Real and complex analysis. Only later I discovered the excellent book on Pick functions by Donoghue, 1974 Monotone Matrix Functions and Analytic Continuation, Springer.