Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 12B / a / Solution

If $f$ is analytic and bounded by $K$ on $\operatorname{Re}z>0$, then Cauchy's derivative estimate gives
$$
|f'(z)|\leq K/c
$$
on $\operatorname{Re}z>c$. Integrating along line segments gives a Lipschitz bound with the same constant in that smaller half-plane.


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 Incoming links: Lipschitz bound inside a bounded analytic half-plane