Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 208 / 3 / a / Solution

The Gaussian logarithmic Sobolev inequality says that for standard Gaussian $X\in {\mathbb{R}}^n$ and smooth $u$,Apply it to $u=\sqrt{f}$. Since $\nabla \sqrt{f}=\nabla f/(2\sqrt{f})$,This sharp inequality immediately implies the requested weaker bound with constant $2$.