One-dimensional Gaussian logarithmic Sobolev inequality: $\mathrm{Ent}_\gamma(g^2)\le 2\int (g')^2\,d\gamma$ for $C^1$ functions $g:\mathbb{R}\to\mathbb{R}$

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One-dimensional Gaussian logarithmic Sobolev inequality: $\mathrm{Ent}_\gamma(g^2)\le 2\int (g')^2\,d\gamma$ for $C^1$ functions $g:\mathbb{R}\to\mathbb{R}$ · Prove2Me