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Review - Handbook of Means and their Inequalities
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Handbook of Means and Their Inequalities


P.S. Bullen

Kluwer Academic Publishers, Dordrecht/Boston/London, 2003, xxvii,

pp. 537, ISBN 1-4020-1522-4

The idea of “mean” is used extensively in Modern Mathematics including Probability Theory & Statistics, Summation of Series and Integrals, in Approximation Theory and related fields.

The main objective of the present book is to provide as complete as possible an account of the properties of means that occur in Theory of Inequalities, and I should say that the author is excellent in this difficult task as many new results following an exponential growth has been discovered, since the previous book devoted to the subject authored by P.S. Bullen, D.S. Mitrinovic and P.M. Vasic and published in late 80ies of the last century.

The introductory chapter is devoted to polynomial properties and some of the basic inequalities needed at various places of the book. They are mostly deduced from the properties of some special polynomials. Some elementary inequalities as well as certain properties of sequences are also presented. A section devoted to convex functions and their inequalities concludes this chapter.

The second chapter is devoted to the properties and inequalities of the classical arithmetic, geometric and harmonic means. In particular the basic inequality between these means, the Geometric Mean-Arithmetic Mean Inequality is discussed. Various refinements of this inequality are then considered; in particular the Rado-Popoviciu type inequalities and the Nanjundiah inequalities. Converse inequalities are discussed as well as Cebysev’s inequality. Some simple properties of the logarithmic and identric means are obtained.

Chapter III is devoted to the properties and inequalities of the classical generalization of the arithmetic, geometric and harmonic means, the power means. The inequalities obtained in the previous chapter are extended to this scale of means. In addition some results for sums of powers are obtained. The classical inequalities of Minkowski, Cauchy and Holder, and some generalization of these results are also mentioned. Various generalizations of the power mean family are pointed out as well.

The fourth chapter is devoted to Quasi-arithmetic means. In this chapter means are defined using arbitrary convex and concave functions by a natural extension of the classical definitions and analogues of the basic results of the earlier chapters are investigated. The generalizations of the geometric-arithmetic and the “(r;s)” inequalities, their converses and the Rado-Popoviciu type extensions are studied under the topic of comparable means.

Chapter V is entirely devoted to symmetric polynomial means. The elementary and complete symmetric polynomials have a history that goes back to Newton. They are used to define means that generalize the geometric and arithmetic means in a complete different way to the above generalizations. In this chapter the author study the properties of these means. Generalizations of these means due to Whiteley and Muirhead are also investigated.

The last chapter includes a variety of topics that do not fit into the previous ones. In particular, there are means that are defined for pairs of numbers and do not readily generalize to n-tuples. There is an elementary introduction to integral means and to matrix analogues of mean inequalities. The topic of axiomatization of means is also briefly discussed.

A large bibliography containing most of the significative papers devoted to the domain followed by a name index and a subject index concludes this interesting book.

The book is dense, with many fundamental result completely proved. In this way, I believe, it is very useful for researchers and postgraduate students that want to develop the domain or use the results in different applications in Probability, Statistics, Numerical Approximations or other domains where means and their inequalities are applied.

In conclusion, I would like to point out that, this excellent book should be on the desks of every mathematician interested in inequalities and their applications.

Sever S. Dragomir

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