Sub-super-stabilizability of certain bivariate means via mean-convexity

Mustapha Raïssouli,József Sándor

doi:10.1186/s13660-016-1212-z

Abstract

In this paper, we first show that the first Seiffert mean P is concave whereas the second Seiffert mean T and the Neuman-Sandor mean NS are convex. As applications, we establish the sub-stabilizability/super-stabilizability of certain bivariate means. Open problems are derived as well.

Highlights

A mean m is a binary map from (, ∞) × (, ∞) into (, ∞) satisfying min(x, y) ≤ m(x, y) ≤ max(x, y) for all x, y >
Symmetric means are defined in the usual way; see, for instance, [ ]
We identify a mean m with its value at (x, y) by setting m = m(x, y) for simplicity

Summary

Introduction

A (bivariate) mean m is a binary map from ( , ∞) × ( , ∞) into ( , ∞) satisfying min(x, y) ≤ m(x, y) ≤ max(x, y) for all x, y >. The super-stabilizability of m is defined in an analogous manner (by reversing the previous mean inequalities). Section contains some applications of the previous results to the sub/super stabilizability of certain bivariate means. If m is symmetric homogeneous, m is partially convex Concave) if and only if the real function x −→ m(x, ) is convex For a mean of class C (i.e., twice differentiable on ( , ∞) × ( , ∞) with continuous second derivative), we can give a direct proof of the previous lemma without refiring to the general result of [ ] stated for general positively homogeneous functions. Such a mean is known as the power (binomial) mean of order p

It is easy to see that

We have φ

According to the explicit form of

It follows that

The best satisfying