TITLE: What is ML Inequality property of complex integral
QUESTION [11 upvotes]: What is ML inequality property in complex integral which says $|\int_{c}f(z)dz| \leq ML$. I can't understand a thing from this expression.
I want to understand it conceptually(if that helps). How can we find the upper bound of a complex integral

REPLY [18 votes]: $$\left|\int_c f(z) \, dz \right| \leq \int_c |f(z)| \cdot |dz|$$
Now assume that $|f(z)| \leq M$ that means the function is bounded on the curve 
$$\int_c |f(z)| \cdot |dz| \leq M \int_c |dz|$$
Now assume that the $c=\gamma(t)$ is a parametrization of the curve  then 
$$\int_c |dz| = \int^b_a |\gamma'(t)| \, dt$$
Now by the definition of the length $L$ of a curve we have 
$$\int^b_a |\gamma'(t)| \, dt=L$$