On the Jensen convex and Jensen concave envelopes of means

TL;DR

This paper investigates the convexity properties of quasiarithmetic means, establishing that their convexity or concavity implies twice continuous differentiability of their generators, and characterizes means with specific Jensen envelopes.

Contribution

It proves that convexity or concavity of quasiarithmetic means implies twice differentiability of their generators, and characterizes means with Jensen convex and concave envelopes.

Findings

Convexity or concavity implies twice differentiability of generators.

Characterization of quasiarithmetic means with Jensen envelopes.

Extension of previous convexity results without differentiability assumptions.

Abstract

In recent papers the convexity of quasiarithmetic means was characterized under twice differentiability assumptions. One of the main goals of this paper is to show that the convexity or concavity of a quasiarithmetic mean implies the the twice continuous differentiability of its generator. As a consequence of this result, we can characterize those quasiarithmetic means which admit a lower convex and upper concave quasiarithmetic envelop.