What is the ML inequality as used in this complex-analysis lecture?

Conditions

• The function is integrated along a rectifiable path.
• The function is continuous on the path.

Reasoning, step by step

1. Identify the function to be integrated, $g(w)$, and the path of integration, $\gamma$.
2. Find the supremum of the absolute value of the function on the path: $\sup_{w \in \gamma} |g(w)|$.
3. Determine the length of the path: $\text{length}(\gamma)$.
4. Multiply the supremum by the length to get the upper bound for the absolute value of the integral: $|\int_{\gamma} g(w) dw| \le \sup_{w \in \gamma} |g(w)| \cdot \text{length}(\gamma)$.
5. Apply this inequality to bound the error term in the difference quotient of the antiderivative.

Example

The lecturer says, 'use the ML inequality,' then explains that 'the integral must be smaller than the supremum of the values taken on the interval you are integrating over... times the length of the interval.' The board shows $\le \frac{1}{|z-a|} \sup_{w \in [a,z]} |f(w)-f(a)| \cdot |z-a|$.

Common misconceptions

• Confusing the ML inequality with the Mean Value Theorem.
• Assuming the inequality provides an exact value for the integral rather than an upper bound.

Watch the explanation

Explore next

• How is the estimation lemma applied to bound the integral of $f(w) - f(a)$?
• Why do the factors 1/|z-a| and |z-a| cancel in the displayed estimate?