In my talk I will introduce a family of functional inequalities introduced recently by myself, Kentaro Fujie and Tatsuya Hosono, that are 'nonlinear extensions' of the inequality between Hessians I talked about at the seminar some months ago. I will emphasize the meaning of the magnitude of the best constant in such inequalities in context of new estimates in PDEs as well as information theory.

Finally, I will show two proofs of the family of such inequalities, the second yielding a better (but not yet sufficient:>) constant.