Are analytic functions injective?

Question

Let $f$ be analytic on the whole all of $\mathbb C$. Assume that $\mathrm{Re}\, f \ge 0$. What can we say about $f$?

I'm thinking $f$ has got to be constant, since otherwise it would map the entire complex plane onto the positive real half-plane. I don't think an analytic function would be capable of this. But I'm not sure how to state it rigorously, if it's even true at all.

I suspect that such a mapping would be 2-1, meaning that $f(z)$ isn't uniquely determined. Is this true/rigorous enough?

Answer 1

The question in the title and in the main body are different. An analytic function need not be injective, just look at $z^2$ on the real axis. However as for the question in the main body, the real part of a holomorphic function is harmonic, and harmonic functions that are bounded below are constant. Since the real part is constant, the Cauchy-Riemann equations imply that the function is constant.

Answer 2

Consider the function

$$g(z) = \frac{1}{f(z) + 1}$$

This is an analytic function (why?) which is bounded (why?).

---

To answer the question in the title, analytic functions are not always injective: $e^z$ is certainly not, nor is $z^2$.