Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 208 / 3

For standard Gaussian measure $\gamma$, the Gaussian Poincaré inequality isThe Gaussian logarithmic Sobolev inequality iswhere ${\mathrm{Ent}}_{\gamma }(G)=\int G\log Gd\gamma -(\int Gd\gamma )\log \int Gd\gamma$.

Set $h=\sqrt{f/\varphi }$. Then $\int h^{2}d\gamma =1$ andMoreover,The Gaussian logarithmic Sobolev inequality therefore gives

With the same $h$, the squared Hellinger distance isSince $0\le \int hd\gamma \le 1$,The Gaussian Poincaré inequality and the derivative calculation in part (b) yield