Complex analysis,complex integration.

Question

Let $f(x)$ be analytical function in convex,open,connected set $\Omega$, and $Re(f(x)) \geq m > 0$.Prove that $\forall x_{1},x_{2} \in \Omega $ holds inequality: $$ \left|\int_{x_{1}}^{x_{2}} f(x)dx\right| \geq m\left|x_{1}-x_{2}\right|$$ I tried to parameterize straigth line $ [x_{1},x_{2}]$ but don't know how to continue.

Answer 1

Hint: Let be $c\in\Bbb C$ s.t. $|c| = 1$ and $c(z_1 - z_2)\in\Bbb R$. Then, $$ \left|\int_{z_{1}}^{z_{2}}f(z)\,dz\right| = \left|c\int_{z_{1}}^{z_{2}}f(z)\,dz\right| = \left|\int_{z_{1}}^{z_{2}}f(z)c\,dz\right|. $$ Can you continue?

Edit: Write $f = u + iv$ with $u\ge M.$ the obvious parametrization $z = z_1 + t(z_2 - z_1)$, $t\in[0,1]$ gives $dz = (z_2 - z_1)\,dt$, i.e. $$f(z)c\,dz = (u(z) + iv(z))c(z_2 - z_1)\,dt$$ An now use the second hypothesis on $c$...