The statement is: Let $\Omega$ be an open convex subset of $\mathbb{C}$, and $f\in\mathcal{H}(\Omega)$ with $|f'(z)-1|<1$, for all $z\in\Omega$. Prove that $f$ is injective.
I want to know if my proof is correct.
Suppose there exist $\alpha,\beta\in\Omega$ with $\alpha\ne\beta$ and $f(\alpha)=f(\beta)$. Because $\Omega$ is convex, we can integrate over the segment $\alpha\beta$, which is in $\Omega$. Then we have $$ \left|\int_{\alpha}^{\beta} (f'(z)-1) \,dz\right| = \left|\left. f(z)-z\right]_{\alpha}^{\beta}\right|= |f(\alpha)-\alpha - f(\beta)+\beta| = |\beta -\alpha|. $$ By the other hand we have $$ \left|\int_{\alpha}^{\beta}( f'(z)-1) \,dz\right|\leq \int_{\alpha}^{\beta}|f'(z)-1||dz| < \int_{\alpha}^{\beta} |dz| = |\beta - \alpha| $$ but this is not possible.
Is this correct? Thanks
