Estimation Lemma for Contour Integrals

Theorem
Let $D \subseteq \C$ be open.

Let $f : D \to \C$ be continuous.

Let $\gamma : [a,b] \to D$ be a path in $D$.

Then:


 * $\displaystyle \left| \int_\gamma f(z)\ dz \right| \leq \int_\gamma |f(z)|\cdot |dz| \leq M L(\gamma) $

where $L(\gamma)$ is the length of $\gamma$:


 * $\displaystyle M = \sup_{z \in \gamma} |f(z)| $

and


 * $\displaystyle \int_\gamma |f(z)|\cdot |dz| = \int_a^b |f(\gamma(t))|\cdot |\gamma'(t)|\ dt$

Proof
We have