ML inequality

The ML inequality is an estimation technique for the upper bound of a contour integral. It tells us that:

$$|{\int }_{C}g(z)dz|\le M\cdot L$$

where $L=b-a$, i.e., the length of $C$. $|g(z)|\le M$ on $C$.

There’s no real perfect bound here: we can go pretty high. But what we do get is a guaranteed upper bound (and not an exact estimate).

Example

Say we want to find an upper bound for:

$${\int }_C\frac{e^z(z+1)}{z^2+4}dz$$

where $C$ is the right-half of a circle where $|z|=5$. If we take $g(z)$ as the integrand, then we’re looking for:

$$M\ge |g(z)|=\frac{|e^z||z+2|}{|z^2+4|}$$

Since

$$|e^z|=e^x$$ $$|e^x|\le e^5$$ $$|z+2|\le 7$$

Then we look at the denominator. We need the denominator term to be an upper bound:

$$|z^2+4|\ge |z^2|-|-4|=25-4=21$$

Then we put everything together.