Sharp informational inequalities involving Kullback-Leibler and Rényi divergences and a family of scaling-invariant relative Fisher measures

We introduce a new transformation called relative differential-escort, which extends the usual differential-escort transformation by relating the change of variable to a reference probability density. As an application of it, we define a biparametric family of relative Fisher measures presenting significant advantages with respect to the pre-existing ones in the literature: invariance under scaling changes and, consequently, sharp inequalities between the new relative Fisher measure and the well established Kullback-Leibler and Rényi divergences. We also introduce a biparametric family of relative cumulative moment-like measures and we establish sharp lower bounds of these new measures by the Kullback-Leibler and Rényi divergences. The optimal bound and the minimizing densities are given. We also construct a family of inequalities for an arbitrary and fixed minimizing density in which the so-called generalized trigonometric functions plays a central role, providing thus one more interesting application of the newly introduced inequalities and measures.