Solution
= Solution
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 $2$.