 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-2/12b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Let

∣ f (z) ∣ \leq K

Re z > 0

. If

Re z > c

∣ w - z ∣ = c

lies in that half-plane. The Cauchy derivative formula gives

∣ f^{'} (z) ∣ \leq \frac{K}{c} .

The line segment joining

z_{1}

z_{2}

remains in

Re z > c

, so

∣ f (z_{1}) - f (z_{2}) ∣ \leq \frac{K}{c} ∣ z_{1} - z_{2} ∣.

Thus one may take $M = K / c$ , as in the lipschitz bound inside a bounded analytic half-plane.

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