Thomas Simpson and the arithmetic mean

The Arithmetic - Harmonic Mean

Consider two sequences generated by ",,+ i Mi"„<hn)hn*\ M'i"„+X,b„), where the a„ and b„ are positive and M and M' are means. The paper discusses the nine processes which arise by restricting the choice of M and M' to the arithmetic, geometric and harmonic means, one case being that used by Archimedes to estimate it. Most of the paper is devoted to the arithmetic-harmonic mean, whose limit is e...

Generalizing the Arithmetic Geometric Mean

The paper discusses the asymptotic behavior of generalizations of the Gauss’s arithmetic-geometric mean, associated with the names Meissel (1875) and Borchardt (1876). The "hapless computer experiment" in the title refers to the fact that the author at an earlier stage thought that one had genuine asymptotic formulae but it is now shown that in general "fluctuations" are present. However, no ve...

Some More Inequalities for Arithmetic Mean, Harmonic Mean and Variance

We derive bounds on the variance of a random variable in terms of its arithmetic and harmonic means. Both discrete and continuous cases are considered, and an operator version is obtained. Some refinements of the Kantorovich inequality are obtained. Bounds for the largest and smallest eigenvalues of a positive definite matrix are also obtained.

The matrix arithmetic-geometric mean inequality revisited

An Arithmetic and Geometric Mean Invariant

A positive real interval, [a, b] can be partitioned into sub-intervals such that sub-interval widths divided by sub-interval ”‘average”’ values remains constant. That both Arithmetic Mean and Geometric Mean ”‘average”’ values produce constant ratios for the same log scale is the stated invariance proved in this short note. The continuous analog is briefly considered and shown to have similar pr...