Must every holomorphic function $f:D(0,1)\longrightarrow D(0,1)$ have a fixed point?
I know that any holomorphic function with two fixed points is the identity: $f=Id$, but I can't find out an holomorphic function without a fixed point.
Appreciate any suggestion.
Best Answer
It depends on the precise definition of $D(0,1)$. If this is open disk then the answer is negative. Indeed, open disk $D$ is conformal to the upper half plane $U$, say, via the map $g: D\to U$. In the latter take the map $f(z)=z+1$ which has no fixed points in the half-plane. Thus, the holomorphic self-map $F=g^{-1}\circ f\circ g $ of $D$ has no fixed points in $D$ either.
If by a disk you mean a closed disk then every continuous self-map of the closed disk (of any dimension) has a fixed point. This is Brouwer's fixed point theorem.