Suppose that $U$ is simply connected domain in $\mathbb{C}$, and let $P$ be a point in $U$. Prove that for all $t \in \mathbb{R}$, there exists a unique holomorphic function $f:U \to U$ such that $f(P)=P$ and $f'(P)=e^{it}$.
Some of my ideas: this is a problem from 2011 UCLA PhD qualifications. Seeing the unique, I want to use the automorphism of the unit disk or Riemann mapping theorem, but I can not construct it. Thanks for your answer very much.
Best Answer
By Riemman Mapping Theorm,there exists a unique conformal map $\phi$:$\mathbb{U}$ $\rightarrow$ $\mathbb{D}$
where $\phi$(P)=0,thus for all t$\in$ $\mathbb{R}$,denote h=$e^it$,consider the map: {g=$\phi$$\circ$h$\circ$$\phi^-1$}\ then:g(P)=P and g'(P)=$\phi$'(0) *$e^it$ * 1/$\phi$'(0)=$e^it$ which satisfies. Thanks for all of your hints!