 Sharpness of the Gaussian logarithmic Sobolev constant

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.