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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