Sharpness of the Gaussian logarithmic Sobolev constant
= Sharpness of the Gaussian logarithmic Sobolev constant
{title2=$g_s(x)=e^{sx/2},\quad \operatorname{Ent}_\gamma(g_s^2)=2\mathcal E_\gamma(g_s)$}
For standard one-dimensional <Gaussian measure>, $\int e^{sx}\,d\gamma=e^{s^2/2}$. Thus $\operatorname{Ent}_\gamma(e^{sx})=(s^2/2)e^{s^2/2}$, while $\int|(e^{sx/2})'|^2d\gamma=(s^2/4)e^{s^2/2}$. Every nonzero $s$ therefore attains ratio two, proving optimality of the Gaussian constant.