Refined convex Sobolev inequalities

This paper is devoted to refinements of convex Sobolev inequalities in the case of power law relative entropies: a nonlinear entropy–entropy production relation im-proves the known inequalities of this type. The corresponding generalized Poincaré type inequalities with weights are derived. Optimal constants are compared to the usual Poincare ́ constant. Key words and phrases: Sobolev inequality – Poincare ́ inequality – entropy method – diffusion – logarithmic Sobolev inequality – spectral ga