 Solution

ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2017/3/a/ii/solution

For

ϕ (t) = r e^{i t}

ϕ^{'} (t) = i r e^{i t}

and

∣ ϕ^{'} (t) ∣ = r

\int_{ϕ} f (z) d z = \int_{0}^{2 π} f (r e^{i t}) i r e^{i t} d t \leq r \int_{0}^{2 π} ∣ f (r e^{i t}) ∣ d t .

This is the circular-contour instance of the estimation lemma.

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