How is the estimation lemma applied to bound the integral of $f(w) - f(a)$?

Conditions

• The path $\gamma_z$ is a rectifiable contour (specifically, a straight line segment).
• The function f is continuous on $\gamma_z$.
• The integrand is $f(w) - f(a)$.

Reasoning, step by step

1. Identify the integral to be bounded: $\int_{\gamma_z} (f(w) - f(a)) dw$.
2. State the estimation lemma: $|\int_{\gamma} g(w) dw| \le \sup_{w \in \gamma} |g(w)| \cdot \text{length}(\gamma)$.
3. Substitute $g(w) = f(w) - f(a)$ and $\gamma = \gamma_z$.
4. Determine the length of the path $\gamma_z$, which is the distance between a and z, i.e., $|z - a|$.
5. Apply the lemma to get the bound: $\sup_{w \in \gamma_z} |f(w) - f(a)| \cdot |z - a|$.
6. Divide by $|z - a|$ (from the difference quotient) to simplify the expression to $\sup_{w \in \gamma_z} |f(w) - f(a)|$.

Example

The board shows the inequality $\le \frac{1}{|z-a|} \sup_{w \in \gamma_z} |f(w) - f(a)| \cdot |z-a|$, and the lecturer explains, '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.'

Common misconceptions

• Forgetting to multiply the supremum by the length of the path.
• Assuming the estimation lemma applies to functions that are not continuous on the path of integration.

Watch the explanation

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• Why can we use a straight line segment to calculate $F(z) - F(a)$ in the difference quotient?
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