Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 208 / 2 / a

Table of contents
- Solution a

Solution

Apply the Poincaré inequality in probability theory to

g = e^{λ f /2}

. Since

∥ \nabla f ∥ \leq 1

,

F (λ) - F (λ /2)^{2} \leq C_{P} (X) \frac{λ ^{2}}{4} F (λ) .

Hence

F (λ) \leq (1 - \frac{λ ^{2} C _{P} ( X )}{4})^{- 1} F (λ /2)^{2} .

Iterating this estimate

m

times yields

F (λ) \leq \prod_{k = 0}^{m - 1} (1 - \frac{λ ^{2} C _{P} ( X )}{4 ^{k + 1}})^{- 2^{k}} F (λ / 2^{m})^{2^{m}},

valid under the stated bound on

λ

.

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