On the homogenization of means

Abstract

The aim of this paper is to introduce several notions of homogenization in various classes of weighted means, which include quasiarithmetic and semideviation means. In general, the homogenization is an operator which attaches a homogeneous mean to a given one. Our results show that, under some regularity or convexity assumptions, the homogenization of quasiarithmetic means are power means, and homogenization of semideviation means are homogeneous semideviation means. In other results, we characterize the comparison inequality, the Jensen concavity, and Minkowski-and Hölder-type inequalities related to semideviation means.2010 Mathematics Subject Classification. 26D10, 26E60.

On the Jensen convex and Jensen concave envelopes of means

“…Following the ideas of convex embeddings (hulls, cones, and so on), there arises a natural problem: How can we associate a convex (or concave) mean to a given one? A quite similar and comprehensive study related to the homogeneity axiom has been presented recently by the authors [15].…”

Estimating the Hardy Constant of Nonconcave Homogeneous Quasideviation Means

“…All these results use Kedlaya (or Kedlaya-type) properties [5,6] in their background, which unifies their assumptions. In the most natural setting, we assume that a mean is concave, homogeneous, and repetition invariant (then it is also monotone).These assumptions are relaxed for example using homogenizations techniques [15,16], or by replacing repetition invariance by a weaker axiom [19]. However we have not been able to relax the concavity assumption.…”

Estimating the Hardy constant of nonconcave Gini means

“…All these results use Kedlaya (or Kedlaya-type) properties [5,6] in their background, which unifies their assumptions. In the most natural setting, we assume that a mean is concave, homogeneous, and repetition invariant (then it is also monotone).These assumptions are relaxed for example using homogenizations techniques [18,19], or by replacing repetition invariance by a weaker axiom [10]. However we have not been able to relax the concavity assumption.…”