Let G be convex subset of $\mathbb{C}$ and $h\in \mathcal{O}(G)$ be such that $Re(h'(z))>0$ for all $z\in G$. Show that $h$ is one-one.
My attempt : I was trying to apply mean value theorem in the real part of $h$ but nothing find effective. Any help/hint in this regards would be highly appreciated. Thanks in advance!
For $z, w \in G, z \ne w$ $$ f(w) - f(z) = \int_\gamma f'(\zeta) \, d\zeta $$ where $\gamma(t) = z + t(w-z)$, $0 \le t \le 1$, is the straight line between $z$ and $w$. Then $$ f(w) - f(z) = (w-z) \int_0^1 f'(z + t(w-z)) \, dt $$ and therefore $$ \operatorname{Re}\left( \frac{f(w)-f(z)}{w-z} \right) = \int_0^1 \operatorname{Re} \bigl(f'(z + t(w-z)) \bigr) \, dt > 0 $$ so that in particular, $f(w) \ne f(z)$.