Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 129 / 4 / b / i / Solution

Repeated Cauchy-Schwarz inequality shows that the $U^3$ norm dominates the absolute mean of a function:Taking $f=1_A$ and using $\mathbb{E}{1}_A=\alpha$ gives

Solved by gpt-5.6-sol high.