Means in Mathematical Analysis: Bivariate Means

By Gheorghe Toader and Iulia Costin

Means in Mathematical Analysis addresses developments in global analysis, non-linear analysis, and the many problems of associated fields, including dynamical systems, ergodic theory, combinatorics, differential equations, approximation theory, analytic inequalities, functional equations and probability theory. The series comprises highly specialized research monographs written by eminent scientists, handbooks and selected multi-contributor reference works (edited volumes), bringing together an extensive body of information. It deals with the fundamental interplay of nonlinear analysis with other headline domains, particularly geometry and analytic number theory, within the mathematical sciences.

• Reviews double sequences defined with two arbitrary means, aiding digestion, analysis and prospective research
• Provides exact solutions on bounds, inequalities and approximations for researchers interrogating means across physical and statistical problems
• Places the current state of means in mathematical analysis alongside its storied and impressive history

• Language: English
• Publisher: Academic Press
• Release date: Sep 14, 2017
• ISBN: 9780128110812

Gheorghe Toader

Gheorghe Toader was born in Romania in 1945, but deferred research until 1980 when he defended his PhD thesis, and served as Professor at the Department of Mathematics of the Technical University of Cluj-Napoca until retirement. From 1970 he has been a referee of Zentralblatt fur Mathematik and since 2007 was in the Editorial Board of Journal of Mathematical Inequalities. He published more than 70 papers related to the subject of this book. Toader sadly passed away in early 2016.

Book preview

2017

Introduction

Motivation for this book The subjects of interest in this book – means and double sequences – are generally associated with the names of great mathematicians such as Pythagoras, Archimedes, Heron, J.L. Lagrange , and C.F. Gauss, to list but a few of the more famous names. Therefore we have to begin with a short introduction into the history of mathematics. Passing then to the current stage of development in the field of means and double sequences, we, as the authors, will refer to our own results, most of which have been published in Romanian journals with limited distribution and may therefore be less known. This doesn't mean that even one relevant result known to us will be omitted.

Throughout the book, references to papers and books are indicated with the name of the author (authors) followed by the year of publication. If an author (or a group of authors) is (are) present in the list of references with more papers published in the same year, small letters are appended to the corresponding year.

With this book we continue the subject developed in Toader and Toader (2005), passing from Greek means to arbitrary means. By Greek means we refer to the means defined by the Pythagorean school. These were presented by Pappus of Alexandria in his books in the fourth century AD (see Pappus, 1932) and they include six (unnamed) means, along with the four well-known means:

, defined by

, given by

, with the expression

, defined by

Chapter 1 Chapter 1 of the book presents some classical problems involving double sequences. The oldest problem involving double sequences was given by Archimedes of Syracuse (287–212 BC) in his book Measurement of the Circle. The problem he was trying to solve was the evaluation of the number π, defined as the ratio of the perimeter of a circle to its diameter.

the half of the perimeters of the inscribed and circumscribed regular polygons with n sides, respectively, we have

To obtain a good estimation, Archimedes passes from a given n to 2n, proving his famous inequalities

As it was shown in are given, step by step, by the relations

, has the value

In Archimedes' case

thus the common limit is π. In Borwein and Borwein (1983a, 1984), the authors describe another form of a double sequence (related also to the Gaussian arithmetic–geometric mean iteration), that is quadratically convergent to π. Also, Newman (1985) defines an alternate version of the arithmetic–geometric mean.

A second example of double sequences is provided by Heron's method of extracting square roots (as it is given in Bullen, 2003). To compute the geometric root of two numbers a and b, we can define

. Of course, the procedure was used only as an approximation method, the notion of limit being unknown in Heron's time.

The third example is Lagrange's method of determining certain irrational integrals. In Lagrange (1784–1985), for the evaluation of an integral of the form

where N , and

have the same first eleven decimals as those determined in Euler (1782) for the integral

. Later, Gauss was able to represent the arithmetic–geometric mean using an elliptic integral, by

As it is well known, this result is used for numerical evaluation of the elliptic integral.

The arithmetic–geometric mean or some similar defined means were studied in many other papers that were not explicitly mentioned in the book, like Borchardt (1861, 1876), Ciorănescu (1936), Cox (1985), Myrberg (1958a), Borwein and Borwein (1989), Schoenberg (1978) and many others.

Chapter 2 Chapter 2 is devoted mainly to those aspects of the theory of means needed in the study of double sequences. We consider some methods of constructing means, some inequalities between means, and we study some operations with means (as, for example, in Toader and Toader, 2006a). For the study of double sequences (which will follow in the next chapter), the most important notions are those of invariance and of complementariness of means. If

we call the mean P -invariant and the mean N to be the complementary of the mean M with respect to the mean P. In Toader and Toader (2005), by direct computation, there were determined the ninety complementaries of a Greek mean with respect to another. For other special means, some problems of invariance were solved as functional equations. Here, in order to determine the complementary of an arbitrary mean with respect to another in a given family of means, we use the series expansion of means. The computations were performed using the computer algebra program Maple, as it is presented in Heck (2003). Many results regarding the complementariness, for example with respect to weighted Gini means or Stolarsky means, cannot be obtained without the help of computer algebra. We have to solve complicated systems of nonlinear equations, sometimes printed on more pages. Though obtained with the help of a computer, the results can be easily verified. Since the complementary of a mean with respect to another can be a pre-mean (thus a reflexive function), we transpose some properties from means to pre-means. We also study the problem of invariance in some families of means. Thus we determine some triples of means with the property that one of them is invariant with respect to the other two.

Chapter . Therefore we can define double sequences of