Thread: Analytic Function in a domain D

If f(z) is analytic in a domain D and maps D onto a portion of a straight line, then show that f(z) is a constant function in D.

Anyone know how to solve this?

Do you know the open mapping theorem?

Don't really because i just start to learn complex analysis.

In fact the open mapping theorem was a sledgehammer proof. We can write , where $\theta is a fixed real number. So is analytic on the open unit disk, and take only real values. By Cauchy-Riemann equations, what can you conclude?