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 \textit{down-moments} and \textit{cumulative upper-moments}. A remarkable fact is that the down-moments provide, in some cases, an interpolation between the $p$-th moments and the power R\'enyi entropies of a probability density. We establish new and sharp inequalities relating these new functionals to the classical informational measures such as moments, R\'enyi 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.

翻译：本文引入了一个双参数变换族，可视为所谓上变换与下变换的推广。该类新变换使我们能够定义新的信息泛函，我们称之为\\textit{下矩}与\\textit{累积上矩}。一个显著的事实是，下矩在某些情况下提供了概率密度函数的$p$阶矩与幂R\\'enyi熵之间的插值。我们建立了这些新泛函与经典信息测度（如矩、R\\'enyi熵、Shannon熵及Fisher信息测度）之间新的精确不等式，并给出了最优界及极小化密度——后者在某些情况下可表示为广义三角函数形式。此外，我们通过新泛函的商定义了一类统计复杂度测度，并利用上变换与下变换的代数共轭性质证明了其单调性。所有这些性质揭示了泛函不等式间错综复杂的结构。