Probability theory can be used to solve certain entirely deterministic functional equations, using a technique pioneered by Aubrun and Nechita. We give some background on moment generating functions, large deviations, and convex conjugates, then state Aubrun and Nechita’s multiplicative characterization of the p-norms (giving a variant of their proof). We then use this theorem on norms to give a multiplicative characterization of the power means.

We prove four theorems characterizing the unweighted power means among all unweighted means. We then build a tool for converting characterization theorems for unweighted means into characterization theorems for weighted theorems. Using this tool, we deduce four theorems characterizing the weighted power means among all weighted means. The main new feature of the theorems proved in this chapter is that they do not assume continuity.

We introduce two families of deformations of Shannon entropy: the q-logarithmic entropies (also called “Tsallis entropies”) and the Rényi entropies. We explain how the exponentials of the Rényi entropies, called the Hill numbers, convey information about the diversity and structure of an ecological community. We introduce the power means, which lie at the technical heart of this book. We give functional equations characterizing the q-logarithmic entropies on the one hand, and the Hill numbers on the other.