Index of a bivariate mean and applications

Mustapha Raïssouli

doi:10.1186/s13660-016-1156-3

Abstract

Exploring some results of (Raissouli in J. Math. Inequal. 10(1):83-99, 2016) from another point of view, we introduce here some power-operations for (bivariate) means. As application, we construct some classes of means in one or two parameters including some standard means. We also define a law between means which allows us to obtain, among others, a simple relationship involving the three familiar means, namely the first Seiffert mean, the second Seiffert mean, and the Neuman-Sandor mean. At the end, more examples of interest are discussed and open problems are derived as well.

Highlights

By a mean we understand a map m between positive real numbers satisfying the following double inequality:∀a, b > min(a, b) ≤ m(a, b) ≤ max(a, b).As usual, continuous means are defined in the habitual way
Continuous means are defined in the habitual way
We identify a mean m with its value at (a, b) by setting m := m(a, b) for the sake of simplicity

Summary

Introduction

By a (bivariate) mean we understand a map m between positive real numbers satisfying the following double inequality:. ). Theorem A Let m be a continuous homogeneous symmetric mean. A continuous homogeneous symmetric mean will be called regular mean, for the sake of simplicity. A regular mean m will be called σ -regular if the map x −→ m(x, ) is continuously differentiable on ( , ∞) and the function fm (called the generated function of m) defined by d x– fm(x) = dx m(x, ) for all x > , with fm( ) = , satisfies the double inequality min , /x ≤ fm(x) ≤ max , /x for all x >. If we denote by Mr and Mσ the sets of all regular means and σ -regular means, respectively, the mean-map m −→ mσ is a bijection from Mr into Mσ and we can write rmσ = m, rm ∈ Mr ⇐⇒ rm = m–σ , m ∈ Mσ.

Then we have

We then deduce that

Let C be defined by

It is clear that

The two means