Suppose $f(z)$ is an entire function, And let $\zeta$ be jordan arc s.t endpoints of the arc $z_1=0$ and $z_2=\infty$. Prove that function is constant if $f(\mathbb{C})\cap\zeta$=empty
I thought this was obvious, but after thinking for a while i need some clarifications:
1.by louvilles theorem entire+bounded=constant. Jordan arc is the image of bounded and closed set, so the image is also bounded and closed under continuous map.Then conditions of louville satisfied.
2.but the arc takes an infinite value. Since the intersection of the arc with complex plane under mapping is empty. It means the interval of real line was mapped to the arc.
3.now how to proceed and show that f is constant?
Picard's Theorem says that that the range of any non-constant entire function is all of $\mathbb C$ or $\mathbb C \setminus \{c\}$ for some complex number $c$. Since your arc has more than one point the result follows.
Ref.: https://en.wikipedia.org/wiki/Picard_theorem