Are analytic functions injective? 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?

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 $z^2$.
