Through and beyond moments, entropies and Fisher information measures: new informational functionals and inequalities

We introduce new classes of informational functionals, called \emph{upper moments}, respectively \emph{down-Fisher measures}, obtained by applying classical functionals such as $p$-moments and the Fisher information to the recently introduced up or down transformed probability density functions. We extend some of the the most important informational inequalities to our new functionals and establish optimal constants and minimizers for them. In particular, we highlight that, under certain constraints, the generalized Beta probability density maximizes (or minimizes) the upper-moments when the moment is fixed. Moreover, we apply these structured inequalities to systematically establish new and sharp upper bounds for the main classical informational products such as moment-entropy, Stam, or Cramér-Rao like products under certain regularity conditions. Other relevant properties, such as regularity under scaling changes or monotonicity with respect to the parameter, are studied. Applications to related problems to the Hausdorff moment problem are also given.