I have already proved Estimation lemma when the complex integration is taken over a smooth curve but I find difficulty to prove it when the complex integration is taken over a piecewise smooth curve i.e. over a contour.
Please help me in proving this.
Thank you in advance.
EDIT $:$
Actually I find difficulty to prove that
$$\left|\int_{\gamma} f(z)\ dz \right| \leq \int_{\gamma} |f(z)|\ |dz|.$$
if $\gamma$ is a contour instead of a smooth curve.
My attempt $:$
Let $\gamma$ having parametric interval $[a,b]$ be composed of finite number of smooth curves say $\gamma = \gamma_1 + \gamma_2 + \cdots + \gamma_n$ where each $\gamma_j$ is smooth in it's own parametric interval $[t_{j-1} , t_j]$ for $1 \leq j \leq n$, where $t_0 = a$ and $t_n = b$.Assuming that $f$ is continuous on $\{\gamma \}$.Let $f(\gamma (t)) \leq M$ for all $t \in [a,b]$.Then
$$\left | \int_{\gamma} f(z)\ dz \right | = \left | \sum_{k=1}^{n} \int_{t_{k-1}}^{t_k} f(\gamma_k (t)) {\gamma_k}' (t)\ dt \right | \leq \sum_{k=1}^{n} \left | \int_{t_{k-1}}^{t_k} f(\gamma_k (t)) {\gamma_k}' (t)\ dt \right | \leq \sum_{k=1}^{n} \int_{t_{k-1}}^{t_k} \left | f(\gamma_k (t)) \right | \left | {\gamma_k}' (t) \right |\ dt \leq M \sum_{k=1}^{n} L(\gamma_k) = ML(\gamma).$$
Which proves the required Estimation lemma.