  We prove a one-parameter family of diffusion hypercontractivity and present
the associated Log-Sobolev, Poincare and Talagrand inequalities. A mean-field
type Bakry-Emery iterative calculus and volume measure based integration
formula (Yano's formula) are presented. Our results are based on the
interpolation among divergence functional, generalized diffusion process, and
generalized optimal transport metric. As a result, an inequality among Pearson
divergence (P), negative Sobolev metric H^-1 and generalized Fisher information
functional (I), named PH^{-1}I inequality, is derived.
