The <Gaussian logarithmic Sobolev inequality> says that for standard Gaussian $X\in\mathbb R^n$ and smooth $u$,
$$
\operatorname{Ent}(u(X)^2)\leq2\mathbb E\lVert\nabla u(X)\rVert^2.
$$
Apply it to $u=\sqrt f$. Since $\nabla\sqrt f=\nabla f/(2\sqrt f)$,
$$
\operatorname{Ent}(f(X))
\leq\frac12\mathbb E\left[\frac{\lVert\nabla f(X)\rVert^2}{f(X)}\right].
$$
This sharp inequality immediately implies the requested weaker bound with constant $2$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-208/3/a/solution