The extremal values of the ratio of differences of power mean, arithmetic mean, and geometric mean

TL;DR

This paper investigates the extremal ratios of differences between power, arithmetic, and geometric means of n variables, providing generalized results and new optimization techniques that extend previous findings.

Contribution

It introduces a novel optimization approach that reduces n-variable problems to a single variable, generalizing and completing earlier specific cases.

Findings

Derived extremal ratios for various means and variables

Established best constant inequalities involving means

Analyzed monotonicity and convergence of constants

Abstract

In the paper the maximum and the minimum of the ratio of the difference of the arithmetic mean and the geometric mean, and the difference of the power mean and the geometric mean of $n$ variables, are studied. A new optimization argument was used which reduces $n$ variable optimization problem to a single variable. All possible cases of the choice of the power mean and the choice of the number of variables of the means are studied. The obtained results generalize and complete the earlier results which were either for specific intervals of power means or for small number of variables of the means. Some of the results are formulated as the best constant inequalities involving interpolation of the arithmetic mean and the geometric mean. The monotonicity and convergence of these best constants are also studied.