What is ML Inequality property of complex integral

Solution 1:

$$\left|\int_c f(z) \, dz \right| \leq \int_c |f(z)| \cdot |dz|$$

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

$$\int^b_a |\gamma'(t)| \, dt=L$$

Solution 2:

$L$ is the arc length of $c$, $M$ is an upper bound for the absolute value of $f$ on $c$.

Let's compare the result to real integrals: Let $f$ be defined on $[a,b]$ and $|f|$ bounded by $M$. Then:

$\left| \int_a^b f(x) dx \right| \leq \int_a^b |f(x)| dx \leq \int_a^b M dx = M \int_a^b dx = M(b-a)$.

Here, $b-a$ is the arc length of $[a,b]$ and $M$ is an upper bound for $|f|$. The ML inequality is just a generalization of this to complex integrals.