  We provide deficit estimates for Nelson's hypercontractivity inequality, the
logarithmic Sobolev inequality, and Talagrand's transportation cost inequality
under the restriction that the inputs are semi-log-subharmonic,
semi-log-convex, or semi-log-concave. In particular, our result on the
logarithmic Sobolev inequality complements a recently obtained result by Eldan,
Lehec and Shenfeld concerning a deficit estimate for inputs with small
covariance. Similarly, our result on Talagrand's transportation cost inequality
complements and, for a large class of semi-log-concave inputs, improves a
deficit estimate recently proved by Mikulincer. Our deficit estimates for
hypercontractivity will be obtained by using a flow monotonicity scheme built
on the Fokker--Planck equation, and our deficit estimates for the logarithmic
Sobolev inequality will be derived as a corollary. For Talagrand's inequality,
we use an optimal transportation argument. An appealing feature of our
framework is robustness and this allows us to derive deficit estimates for the
hypercontracivity inequality associated with the Hamilton--Jacobi equation, the
Poincar\'e inequality, and for Beckner's inequality.
