Consider the integral:
$I=\int_0^zw dw$
We have $|I|\le \int_0^z|w| dw \le \int_0^z|z| dw \le |z| \int_0^z dw $
The last inequality doesn't make sense for me because $\int_0^z dw$ is not real and I can't see my mistake.
Could you please help?
Thank you!
Informally, $dw$ is in general not real, so the first inequality is where the problem is. More formally, parameterising the contour by $w=g(t)$ means that $g'(t)$ is not real. One sometimes writes $$ \left\lvert \int_a^b f(z) \, dz \right\rvert \leq \int_a^b \lvert f(z) \rvert \lvert dz \rvert $$ for the inequality you're looking for, which means that when the contour is parametrised, the integral becomes $$ \int_a^b \lvert f(z) \rvert \lvert dz \rvert = \int_{g^{-1}(a)}^{g^{-1}(b)} \lvert f(g(t)) \rvert \lvert g'(t) \rvert \, dt. $$