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Means in Mathematical Analysis: Bivariate Means

Gheorghe Toader and Iulia Costin
Academic Press
Publication Date: 
Number of Pages: 
Mathematical Analysis and Its Applications
[Reviewed by
Allen Stenger
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This is a very specialized monograph, on two subjects: (1) means of two variables; (2) sequences of pairs of numbers (“double sequences”) defined by a recurrence using a mean. Gheorghe Toader (1945–2015) was a Romanian mathematician who worked in means, inequalities, and other areas of classical analysis. His wife, Silvia Toader, is also a mathematician, and his daughter Iulia Costin is a computer scientist, and Costin prepared this posthumous work for publication.

A simple and ancient example is Heron’s method for finding square roots. To find \(\sqrt{x}\), we pick two numbers \(a_0\), \(b_0\) such that \(a_0 b_0 = x\) and then calculate repeatedly \[a_{n+1} = \frac{2 a_n b_n}{a_n + b_n}, \qquad\qquad b_{n+1} = \frac{a_n + b_n}{2}.\] (The expression for \(a_{n+1}\) is the harmonic mean, and for \(b_{n+1}\) is the arithmetic mean.) These sequences satisfy the invariant \(a_n b_n = x\), and the two sequences \(a_n\) and \(b_n\) tend to a common limit, which is therefore \(\sqrt{x}\). The arithmetic-geometric mean (AGM), pioneered by Lagrange and Gauss in the context of elliptic integrals, is defined similarly, with the harmonic mean replaced by the geometric mean.

The ancient Greeks defined ten kinds of means, including the familiar arithmetic mean and geometric mean, and many more means have been invented since then. Much is this book is devoted to means. It develops some general properties of means as well as properties of many specific means that are useful with double sequences. It then develops properties of double sequences, especially speed of convergence. This is done both for general sequences and for particular choices of mean.

Other books with some overlap with this one include Bullen’s Handbook of Means and Their Inequalities (about means but not about sequences) and Borwein & Borwein’s Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (about many subjects, with the AGM being a central thread; Chapter 8 is about means).

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.


1. Classical Theory of the AGM

2. Means

3. Double Sequences

4. Integral Means


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