We say a map $f:\mathbb C \to \mathbb C$ is contraction(Lipschitz) if $|f(z_{1})- f(z_{2})| \leq C |z_{1}- z_{2}|$ for every $z_{1}, z_{2} \in \mathbb C$ and $C$ is some constant.
Trivial Examples: (a) $f(z)= z, (z\in \mathbb C )$ (b) $f(z)= |z|, (z\in \mathbb C).$
My Question is: (1) What other examples of contraction maps on $\mathbb C$ one can think of ? (2) Can we characterize contraction maps on $\mathbb C$ ?
Thanks,
If $f\colon\mathbb{C}\to\mathbb{C}$ is Lipschitz with constant $C$, then for all $z\in\mathbb{C}$ $$ |f(z)|=|f(z)-f(0)+f(0)|\le C\,|z|+|f(0)|. $$ If $f$ is also (complex) differentiable (that is, it is an entire function), it follows from Liouville's theorem that $f(z)=A\,z+B$ for some $A,B\in\mathbb{C}$.
If no (complex) differentiability condition is required, there are many others. For instance $f(z)=|z|$ or $f(z)=u(x,y)+v(x,y)\,i$ where $u$ and $v$ have bounded partial derivatives.