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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 $2$.

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