Generalized informational functionals and new monotone measures of statistical complexity

In this paper we introduce a biparametric family of transformations which can be seen as an extension of the so-called up and down transformations. This new class of transformations allows to us to introduce new informational functionals, which we have called $\mathit{\text{down-moments}}$ and $\mathit{\text{cumulative}}\mathit{\text{upper-moments}}$. A remarkable fact is that the down-moments provide, in some cases, an interpolation between the $p$-th moments and the power Rényi entropies of a probability density. We establish new and sharp inequalities relating these new functionals to the classical informational measures such as moments, Rényi and Shannon entropies and Fisher information measures. We also give the optimal bounds as well as the minimizing densities, which are in some cases expressed in terms of the generalized trigonometric functions. We furthermore define new classes of measures of statistical complexity obtained as quotients of the new functionals, and establish monotonicity properties for them through an algebraic conjugation of up and down transformations. All of these properties highlight an intricate structure of functional inequalities.