When do real, analytic, monotonic functions on an interval extend to univalent functions on an open region of the complex plane?

Question

Suppose I have a real, analytic function $f(x)$ which is monotonic on some connected, open subset of the real line $W \subseteq \mathbb{R}$ and such that $f'(x)>0$ on $W$. Let me naturally include this function into the complex plane $f(x) \rightarrow f(z)$. When can I guarantee that there is some open region of the complex plane $D \subseteq \mathbb{C}$ where the function $f(z)$ is univalent and such that $W \subseteq D$?