Sharpness of the Gaussian logarithmic Sobolev constant ($g_s(x)=e^{sx/2},{\mathrm{Ent}}_{\gamma }(g_s^2)=2{\mathcal{E}}_{\gamma }(g_s)$)

For standard one-dimensional Gaussian measure, $\int e^{sx}d\gamma =e^{s^2/2}$. Thus ${\mathrm{Ent}}_{\gamma }(e^{sx})=(s^2/2)e^{s^2/2}$, while $\int |(e^{sx/2}{)}^{\prime }{|}^{2}d\gamma =(s^2/4)e^{s^2/2}$. Every nonzero $s$ therefore attains ratio two, proving optimality of the Gaussian constant.