It was pretty hard finding a short and precise title. Here is my problem:
The equation $\bigg|\int_\gamma f(z)\text{d}z\bigg|\le\int_\gamma\big|f(z)||\text{d}z|$ holds if $f$ is integrable (where $\gamma$ is a differentiable path $[0,1]\rightarrow\mathbb{C}$). However, in what cases does "$=$" apply? Some cases are clear; for example if $f(z)=$const, $\forall z\in {\rm Img}(\gamma)$.
Our second idea was that both sides are equal if (or even iff?) $f(z)$ stays within one quadrant of the complex plane for all points in ${\rm Img}(\gamma)$.
So basically, the question is, when does "$=$" apply and is our second idea correct? Use "if" or "iff" (which is if and only if). Is there a nice proof to whatever answer there is? (No need to print the full proof; the basic idea suffices)
EDIT: instead of using the path definition of the integral, here is the corrected version: $\bigg|\int_0^1f(\gamma(t))\gamma'(t)\text{d}t\bigg|\le\int_0^1|f(\gamma(t))||\gamma'(t)|dt.$