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Book review

Author(s) Mitrinovic, D.S.
Pecaric, J.E.
Fink, A.M.
Title Classical and New Inequalities in Analysis
Publisher Kluwer Academic Publishers
Year of publication 1993
Reviewed by P.P. Zabrejko

This book is augment a remarkable row of books that are devoted to an important field in classical analysis and other 'analytical' branches of mathematics namely to inequalities. Every mathematician knows excellent books by G.H. Hardy, J.E. Littlewood and G. Pólya, E.F. Beckenbach and R. Bellman, D.S. Mitrinovich, numerous collections of articles devoted to this topic but this new book will turn out to be an essential addition to them and will be useful both specialists working in field and all mathematicians as a complete and exhausting reference book in which one can find all main classical inequalities associated with names, Gauss, Cauchy, Schwarz, Buniakowsky, Hölder, Minkowski, Cebyshev, Ljapunov, Gram, Bessel, Hadamard, Landau, Bernstein, Hilbert, Hardy, Littlewood, Pólya, Markoff, Kolmogorov, etc. as well as their recent refinement and generalizations, numerous and various connections between them that have not been noticed before. Of course no book, even such big as this, can not cover all up-to-date field of inequalities and in particular it is not possible to find in it numerous inequalities related with norms or other characteristics of operators, so-called geometrical inequalities and some others, but as for 'usual inequalities with numbers and vectors' the book given is the fullest, most comprehensive and most modern one.

The book contains 30 chapters. Chapter 1 'Convex functions' gives various generalizations of Jensen's inequalities with some small sketch of the convex functions theory; in addition to usual and integral forms of Jensen's inequality the Hermite-Hadamard's, Petrovic and Giaccardi inequalities are considered as well as the main inequalities for means. Chapter 2 'Some recent results involving means' some generalizations of classical inequalities for means, namely, inequalities of Sierpinski, Ky Fan, Levinson Bernoulli's inequality, Cauchy's and many their generalizations. Chapter 3 'Bernoulli inequality' is devoted to Bernoulli's inequality <formula> and its generalizations in complex domain, in multidimensional case, Weierstrass's type inequalities and many other. Chapter 4 'Cauchy's and related inequalities' presents Cauchy's classical inequality and its variants and refinements including inequalities of Ostrowski and Fan-Todd. Chapter 5 'Hölder and Minkowski inequalities' is devoted to Hölder's and Minkowski's inequalities and their modifications and refinements including Ljapunov-'s, Aczél's, Popoviciu and Bellman's ones. Chapter 6 'Generalized Hölder and Minkowski inequalities' gives such generalizations like ones with different positive functionals instead of classical lp-norms, inequalities by Beckenbach and Dresher, Prékopa-Leindler, Brunn-Minkowski-Lusternik ones. Chapter 7 'Connections between general inequalities' is devoted to equivalence problems for inequalities of Cauchy, Minkowski and their modifications and generalizations. All these seven chapters are an unified part of the book; its content cover corresponding parts of other books on inequalities with an high excess.

Chapter 8 'Some determinantal and matrix inequalities' deals with Hadamard's inequality and its variants; numerous applications of presented theory to inequalities that were discussed in previous chapters are collected here too. Chapter 9 'Cebychev's inequality' is very big; it is devoted to Cebyshev inequality and its generalizations, that give lower bounds for scalar production of nonnegative functions; the next chapter 10 'Grüss' inequality' deals with numerous estimates for difference and quotient of both sides in Cebyshev inequality.

Chapter 11 'Steffensen's inequality' presents Steffensen's inequality and its analogues; these inequalities give upper and lower estimates for scalar product of two functions in terms of integral from the first function over intervals which are defined by the second one. Chapter 12 'Abel's and related inequalities' is devoted to linear inequalities and analogy of Jensen's inequality for nondecreasing functions. Chapter 13 'Some inequalities for monotone functions' is devoted to different inequalities with absolutely monotonic and completely monotonic functions. Chapter 14 'Youngs inequality' presents a brief but complete sketch of all main results concerning with Young inequality and its converses. Chapter 15 'Bessel's inequality' gives Bessel's inequality, its generalizations offered by R.P. Boas, E. Bombieri, A. Selberg, H. Heilbronn, numerous Gram type determinantal inequalities, Hausdorff-Young inequality and some others. Chapter 16 'Cyclic inequalities' is devoted to inequalities for symmetric functions in several variables; the simplest amongst them is Schur's one on nonnegativeness of special function of three variables; the main part of chapter deals with interesting trigonometric inequalities; moreover, the well known inequalities by Shapiro, Diananda, Mordell and Daykin, Zulauf and so on are considered; some inequalities with weighted cyclic sums are considered too. Chapter 17 'Triangle inequa]ities' gives a fine analysis of 'triangle inequality' and some close inequalities in different concrete metric and normed spaces. Two next chapters 18 and 19, 'Norm inequalities' and 'More on norm inequalities', continue the topic of norm inequalities; the first of them deals with Clarkson's, Dunkl-Williams' and Hlawka's ones with variants; the second deals with Khintchine's, Grothendieck's ones with variants and, moreover, gives a lot of trigonometric inequalities. Chapter 20 'Gram's inequality' presents a good survey of inequalities of Gram type.

Chapter 21 'Fejér-Jackson inequalities and related results' deals with different inequalities for trigonometrical polynomials and series for which coefficients form positive and monotonic systems. Chapter 22 'Mathieu's inequality' deals with upper and lower estimates of sum <formula>. Chapter 23 'Shannon's inequality' is a short survey of different approaches to information theory foundations including Rényi's concept of information measure. Chapter 24 'Turan's inequality from the power sum theory' is a sketch for some inequalities on estimates of power series sums offered by P. Turan in his book 'On a New Method of Analysis and it Applications' (New York - Chichester - Brisbane - Toronto - Singapore, 1984). The last chapters 25-30, 'Continued fractions and Páde approximations method', 'Quasilinearization methods for proving inequalities', 'The centroid method in inequalities', 'Dynamic programming and functional equation approaches to inequalities', 'Interpolation inequalities' and 'Convex mini max inequali-ties-equalities' present a row of methods for constructing new inequalities.

One can see that even the simple enumeration of all results included in the book demands a great deal of room and in deed such enumeration is not possible. It is necessary to add that any chapter contains reach a historical essay, the description of numerous relations between different variants of inequalities considered in the book and references. The book is written with simple and intelligible language: it is acceptable to graduate and postgraduate students. So the book is very useful for all specialists in field as a review book of inequalities and their proofs including the most modern; it is suitable as a textbook in field too. But this book is considerably more useful and interesting for mathematicians who are not specialists in inequalities however repeatedly deal with various inequalities as the base instrument of their own investigations.