Transport proofs of some functional inverse Santal{o} inequalities

In this paper, we present a simple proof of a recent result of the second author which establishes that functional inverse-Santal{o} inequalities follow from Entropy-Transport inequalities. Then, using transport arguments together with elementary correlation inequalities, we prove these sharp Entropy-Transport inequalities in dimension 1. We also revisit the proof of the functional inverse-Santal{o} inequalities in the n dimensional unconditional case using these ideas.