A new sharp inequality featuring the differential Rényi entropy, the Rényi divergence and the Rényi cross-entropy of a pair of probability density functions is established. The equality is reached when one of the probability density function is an escort density of the other. This inequality is applied, together with a general framework of a pair of transformations reciprocal to each other, to derive a number of further inequalities involving both classical and new informational functionals. A remarkable fact is that, in all these inequalities, the Rényi divergence of two probability density functions is sharply bounded by quotients of informational functionals of cross-type and single type. More precisely, we derive sharp inequalities composed by relative and cross versions of the absolute moments, or of the Fisher information measures (among others), and involving two and three probability density functions.

翻译：本文建立了关于概率密度函数对的微分Rényi熵、Rényi散度与Rényi交叉熵的精确不等式，其等号成立条件为其中一个概率密度函数是另一个的伴随密度。结合互为逆变换的一般框架，该不等式被应用于推导涉及经典及新型信息泛函的系列不等式。值得注意的是，在所有推导的不等式中，两个概率密度函数的Rényi散度均被交叉型与单一型信息泛函的商严格界定。具体而言，我们建立了涉及两个或三个概率密度函数的精确不等式，这些不等式由绝对矩或Fisher信息测度（及其他量）的相对版本与交叉版本构成。