Reducible means and reducible inequalities

TL;DR

This paper explores the reducibility of means and convexity properties, showing that certain inequalities hold across different variable counts, and introduces a broad class of means with this property, along with conditions for reducibility.

Contribution

It generalizes the concept of reducibility to a wide class of means and convexity notions, providing new sufficient conditions for reducibility of inequalities.

Findings

Established a class of generalized means with reducibility property

Provided sufficient conditions for reducibility of $(M,N)$-convexity

Extended reducibility concepts to H"older--Minkowski inequalities

Abstract

It is well-known that if a real valued function acting on a convex set satisfies the $n$-variable Jensen inequality, for some natural number $n\geq 2$, then, for all $k\in\{1,\dots, n\}$, it fulfills the $k$-variable Jensen inequality as well. In other words, the arithmetic mean and the Jensen inequality (as a convexity property) are both reducible. Motivated by this phenomenon, we investigate this property concerning more general means and convexity notions. We introduce a wide class of means which generalize the well-known means for arbitrary linear spaces and enjoy a so-called reducibility property. Finally, we give a sufficient condition for the reducibility of the $(M,N)$-convexity property of functions and also for H\"older--Minkowski type inequalities.