SUPERADDITIVITY OF FISHER INFORMATION AND LOGARITHMIC SOBOLEV INEQUALITIES

We prove a theorem characterizing Gaussian functions and we prove a strict superaddivity property of the Fisher information. We use these results to determine the cases of equality in the logarithmic Sobolev inequality on Rn equipped with Lebesgue measure and with Gauss measure. We also prove a strengthened form of Gross's logarithmic Sobolev inequality with a "remainder term" added to the left side. Finally we show that the strict form of Gross's inequality is a direct consequence of an inequality due to Blachman and Stam, and that this in turn is a direct consequence of strict superadditivity of the Fisher information. © 1991.