 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-1/23g/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

\int_{0}^{\infty} ∣ G (y) g (y) ∣ d y \leq \int_{0}^{\infty} \int_{0}^{\infty} ∣ F (x, y) g (y) ∣ d x d y = \int_{0}^{\infty} (\int_{0}^{\infty} ∣ F (x, y) g (y) ∣ d y) d x .

y

variable for each fixed

x

yields

\int_{0}^{\infty} ∣ G (y) g (y) ∣ d y \leq \int_{0}^{\infty} (\int_{0}^{\infty} ∣ F (x, y) ∣^{p} d y)^{1/ p} ∥ g ∥_{q} d x,

and

∥ g ∥_{q} \leq 1

gives the claimed inequality. This is the duality proof of the Minkowski integral inequality.

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