The problem here:
$U=\{z|\Im(z)\leq\pi/2\}$, $f$ be an entire function such that $f(U)\subset U, f(-1)=0,f(0)=1$, prove that $f(z)=z+1$.
My idea: I think that the condition of the value of $f$ is just to determine the “parameter”, what we have to do is to prove that $f=az+b$. however, how to use the condition $f(U)\subset(U)$?
Any entire function that preserves a half-plane and has two fixed points there is the identity (map the half-plane to the unit disc by a holomorphic bijection st one point goes to $0$ and use Schwarz lemma).
Using this for $g(z)=f(z)-1$ with two fixed points and still preserving the half-plane $U$ gives the result