Complex Differentiable Functions Mapping Upper Half Plane to Itself? Let $H$ denote the upper half plane of $\mathbb{C}$ (not including the real line).  I have a theorem in my book that states:
"Every automorphism of $H$ is a fractional linear transformation $f(z) = \frac{az + b}{cz + d}$ such that $ad - bc = 1$."
I've been trying to find some other functions (not necessarily bijections) that map $H$ into $H$ and are complex differentiable.  Of course multiplication by a real number is one example, but isn't very interesting.  Polynomials $f(z) = z^n$ don't map $H$ to $H$, although they are differentiable.  The function $f(z) = \frac{1}{\overline{z}}$ maps $H$ to $H$, but isn't differentiable.  Ideas?
 A: The map
$$S \colon z \mapsto i\frac{1+z}{1-z}$$
is a biholomorphism $\mathbb{D}\to \mathbb{H}$, its inverse is
$$S^{-1} \colon z \mapsto \frac{z-i}{z+i}.$$
The conjugation with $S$ therefore induces a bijection $\operatorname{Hol}(\mathbb{H},\mathbb{H}) \to \operatorname{Hol}(\mathbb{D},\mathbb{D})$, $f \mapsto S^{-1}\circ f \circ S$.
Since we have $\lvert z\cdot w\rvert = \lvert z\rvert \cdot \lvert w\rvert$, the product of functions mapping the unit disk to itself is again a function mapping the unit disk to itself, and that gives an easy way to construct more complicated functions mapping the unit disk to itself from easier ones.
A nearly trivial example of functions mapping the unit disk to itself is the power $p_n \colon z \mapsto z^n$ for $n \in \mathbb{Z}^+$. Conjugating that with $S$, we obtain the function mapping the upper half plane to itself
$$S\circ p_n \circ S^{-1} : \colon z \mapsto S\left(\left(\frac{z-i}{z+i}\right)^n\right) = i\frac{(z+i)^n + (z-i)^n}{(z+i)^n-(z-i)^n}$$
which, to be sure, is not obvious.
A further way to obtain interesting functions mapping the upper half plane to itself arises from the observation that $\mathbb{H} + \mathbb{H} = \mathbb{H}$, so the sum of functions mapping the upper half plane to itself is again a function mapping the upper half plane to itself. So adding two automorphisms,
$$z \mapsto \frac{az+b}{cz+d} + \frac{\alpha z+\beta}{\gamma z+\delta} = \frac{(a\gamma + c\alpha)z^2 + (a\delta +b\gamma + c\beta + d\alpha)z + b\delta+\beta d}{(cz+d)(\gamma z+\delta)}$$
again yields a, usually nontrivial, function mapping the upper half plane to itself.
Combining those options yields lots of interesting functions.
