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?
In fact the open mapping theorem was a sledgehammer proof. We can write $\displaystyle $f(z)=e^{i\theta z}a(z)$$, where $\theta is a fixed real number. So $\displaystyle g(z):=e^{-i\theta}f(z)$ is analytic on the open unit disk, and take only real values. By Cauchy-Riemann equations, what can you conclude?