On the Jensen convex and Jensen concave envelopes of means (Q2025601)

The standard convention is that properties of means like (Jensen) convexity, symmetry, continuity, etc. refer to the respective properties of multivariable functions \(M\mid \{I_n\}\). In this spirit, one can easily see that the arithmetic mean is continuous, affine, symmetric, etc. Jensen convex and Jensen concave means are two narrow families which play an important role in the investigation of inequalities involving means, especially the Ingham-Jessen property. Nowadays, the convexity of quasiarithmetic means are characterized under twice differentiability assumptions. The authors main goals of this paper is to show the convexity or concavity of a quasiarithmetic mean which implies the twice continuous differentiability of its generator. As a consequence of this result, they characterize those quasiarithmetic means which admit a lower convex and upper concave quasiarithmetic envelope. The paper gives many ideas for future research.