Example of a complex valued contraction mapping

Question

I'm trying to find an example of a holomorphic function $f:\mathbb{C}\to\mathbb{C}$ that is a contraction mapping, besides the obvious example of $f(z)=az$, for $|a|<1$. Can anyone help me think of one? I initially thought this would be simple, but it continues to elude me.

Answer 1

Notice that $g(z)=\frac{f(z)-f(0)}{z}$ is a bounded meromorphic function on $\Bbb C$. So it is constant.

This tells you that, more in general, an entire function is Lipschitz-continuous if and only if it is in the form $az+b$.