On Seiffert-like means

TL;DR

This paper explores a new representation of homogeneous, symmetric means using a specific functional form, enabling improved comparisons and estimates of these means in relation to known means.

Contribution

It introduces a novel representation for symmetric, homogeneous means, facilitating new methods for comparing and estimating these means relative to known benchmarks.

Findings

Derived optimal bounds for means using the new representation

Provided estimates relating different symmetric means

Enhanced methods for comparing means in mathematical analysis

Abstract

We investigate the representation of homogeneous, symmetric means in the form M(x,y)=\frac{x-y}{2f((x-y)/(x+y))}. This allows for a new approach to comparing means. As an example, we provide optimal estimate of the form (1-\mu)min(x,y)+ \mu max(x,y)<= M(x,y)<= (1-\nu)min(x,y)+ \nu max(x,y) and M((x+y)/2-\mu(x-y)/2,(x+y)/2+\mu(x-y)/2)<= N(x,y)<= M((x+y)/2-\nu(x-y)/2,(x+y)/2+\nu(x-y)/2) for some known means.