##
**Analytic inequalities. In cooperation with P. M. Vasić.**
*(English)*
Zbl 0199.38101

Die Grundlehren der mathematischen Wissenschaften. 165. Berlin: Springer-Verlag (ISBN 978-3-642-99972-7/pbk; 978-3-642-99970-3/ebook). vi, 400 p. (1970).

Until the publication of this volume there were only two major works in English which dealt with inequalities of a fairly general character. These were the books “Inequalities” by G. H. Hardy, J. E. Littlewood, and G. Pólya [2nd ed., Cambridge (1952; Zbl 0047.05302)] hereafter denoted by HLP, and “Inequalities” by E. F. Beckenbach and R. Bellman [2nd rev. printing, Springer (1965; Zbl 0126.28002)] hereafter denoted by BB. The present volume is not only longer than the others (400 pages as compared to 324 and 198 pages, respectively), but contains many more inequalities than either of them. Although there is obviously some overlapping of content, it is true that “Analytic Inequalities” is devoted in large part to topics not included in the other two. Although the reviewer did not actually take a count, many of the inequalities dealt with here have appeared in papers published since 1960, and a certain number of results (not only by Mitrinović or Vasić) appear for the first time in this volume. The adjective “Analytic” in the title of this book should only be taken to indicate that the inequalities included are – like those in HLP – of a general kind, and that various special kinds of inequalities such as those arising in geometry, probability theory, number theory, the theory of orthogonal functions, and the theory of univalent and multivalent functions, have been excluded.

The present volume is quite different in style from HLP. Whereas the latter book contains proofs (or indications of proof) of most of the inequalities which are introduced, probably fewer than a third of the inequalities of “Analytic Inequalities” are proved in the book. In fact the style of this book is perhaps closer to that usually associated with the “Ergebnisse” series (in which BB appears) than to that of the “Grundlehren” series. Unfortunately, one of the consequences of this style (especially in the last third of Part 2, and in Part 3) is the tendency for trivial and deep results to appear almost side-by-side, with little warning to the reader as to the category in which a given result falls.

The book is divided into three parts, and for a brief summary of these parts, we quote from the author’s Preface: “In the first part – “Introduction” – an approach to inequalities is given, while the main attention is devoted to the Section on Convex Functions. The second and probably main part – “General Inequalities” – consists of 27 sections, each of which is dedicated to a class of inequalities of importance in Analysis ... Finally, the third part – “Particular Inequalities” – is aimed at providing a collection of various inequalities, more or less closely interconnected, some of which are of considerable theoretical interest. They are classified in a certain manner, although we must admit that this has not been done perfectly. Part 3 is, in fact, a collection of over 450 special inequalities, and with a few exceptions we were able to add bibliographical references for each one. Owing to lack of space, only a few inequalities are supplied with a complete proof.”

Rather than trying to give a detailed list of the extremely large number of results in this volume, the reviewer believes it would be more useful to comment on some of those results which are not to be found in HLP or in BB. In Part 1 some extensions and generalizations of the concept of convex functions due to I. E. Ovcharenko, M. A. Krasnosel’skiĭ and Ya. B. Rutiskiĭ, and Á. Császár are given, without proof. In Part 2, recent refinements and extensions of such fundamental inequalities as those of Bernoulli, Cauchy, Chebyshev, Hölder, and Minkowski are given.

Section 2.10 deals with inequalities due to J. Aczel, T. Popoviciu, S. Kurepa, and R. Bellman of which the following one of Popoviciu is typical: If \(p\ge 1\), and \(a= (a_1,\ldots,a_n)\) and \(b=(b_1,\ldots,b_n)\) are sequences of nonnegative numbers such that \(A = a_1^p - a_2^p - \ldots a_n^p > 0\) or \(B =b_1^p - b_2^p - \ldots b_n^p > 0\), then \(AB \le (a_1b_1 - a_2b_2 - \ldots - a_nb_n)^p\).

In Section 2.11 certain “inverse” inequalities of Cauchy’s and Hölder’s inequalities due to J. B. Diaz and F. T. Metcalf, and Z. Nehari are given. Although the interesting inequality of G. Grüss dates back to 1935, it appears to have been overlooked in both HLP and BB. This inequality states that if \(f\) and \(g\) are Riemann integrable over \([a,b]\), with \(\varphi \le f(x) \le \Phi\) and \(\gamma\le g(x)\le \Gamma\) for all \(x\), then \[ \left\vert \frac1{b-a} \int_a^b fg\,dx - \frac1{(b-a)^2} \int_a^b f\,dx \int_a^b g \,dx \right\vert \le \frac14 (\Phi -\varphi) (\Gamma - \gamma), \] and the constant \(1/4\) is best possible. This result is proved in Section 2.13, and several generalizations and extensions are stated without proof.

Section 2.18 is devoted to a presentation (without proof) of certain inequalities involving the modulus of the sums of powers of complex numbers, due to P. Turán, introduced in his book “Eine neue Methode in der Analysis and deren Anwendungen.” Budapest: Akadémiai Kiadó (1953; Zbl 0052.04601), and to more recent extensions and improvements of these results by F. V. Atkinson and others.

In Section 2.19 the authors present a new elementary method for proving inequalities involving integrals of functions and their derivatives, due to D. C. Benson [J. Math. Anal. Appl. 17, 292–308 (1967; Zbl 0146.07404)].

A similar presentation of the method of “recurrent inequalities” of R. Redheffer [Prot. Lond. Math. Soc. (3) 17, 683–699 (1967; Zbl 0156.05804)] is given in Section 2.20; this powerful method applies only to discrete inequalities.

One of the longest sections in the book deals with two integral inequalities involving derivatives – Wirtinger’s inequality and Opial’s inequality – and with the numerous generalizations and extensions of these inequalities. (Incidentally, the authors point out that the name Wirtinger should probably not be attached to this inequality because it had been considered somewhat earlier by a number of other authors.)

In Section 2.25 several inequalities for vector norms are given, one of the most interesting and useful being the following reduction theorem of F. W. Levi [Arch. Math. 2, 24–26 (1949/50; Zbl 0034.29604)]: Let \(k\) and \(P_{ij}\) \((i = 1, \ldots,n; j = 1,\ldots, r)\) be real constants. Suppose that for all real numbers \(x_1,\ldots, x_r\) we have \[ \sum_{i=1}^n k_i\, \vert p_{i1}x_1 + \ldots + p_{ir}x_r\vert \ge 0. \] Then \[ \sum_{i=1}^n k_i\, \Vert p_{i1}a_1 + \ldots + p_{ir}a_r\Vert \ge 0 \] holds for arbitrary vectors \(a_1, \ldots,a_r\in E^*\) and any positive integer \(m\).

Part 3 is the longest part of the book (200 pages, compared to 26 and 160 pages for Parts 1 and 2). It contains nine sections bearing the following titles: Inequalities involving functions of discrete variables (11 pp.); Inequalities involving algebraic functions (19 pp.); Inequalities involving polynomials (18 pp.); Inequalities involving trigonometric functions (12 pp.); Inequalities involving trigonometric polynomials (19 pp.); Inequalities involving exponential, logarithmic and gamma functions (23 pp.); Integral inequalities (21 pp.); Inequalities in the complex domain (26 pp.); Miscellaneous inequalities (50 pp.). A surprisingly large number of the inequalities listed in Part 3 are drawn from the problems (and other) sections of such journals as the Mathematical Gazette, Mathematics Magazine, Mathematics Student, Wisk. Opgaven, and especially from the American Mathematical Monthly.

A further indication of the encyclopedic character of this volume can be inferred from the fact that there are over 750 names cited in the Name Index. (An interesting feature of this index is that the dates of birth, or the dates of birth and death, are given for practically all entries.)

Bibliographic references are given at the end of each section or subsection throughout the book. A much higher proportion of references to Slavic sources is given here than in either HLP or BB.

In the reviewer’s opinion this book is a valuable contribution to the field, and should be on the bookshelf of mathematicians, engineers, statisticians, physicists, and all who use inequalities in their work. By and large, it is probably safe to say that if an inequality – or a reference to a kind of inequality – which one requires can not be found in “Analytic Inequalities”, then either it is one of the special kind specifically excluded from this volume as noted above, or it can likely not be found anywhere.

The present volume is quite different in style from HLP. Whereas the latter book contains proofs (or indications of proof) of most of the inequalities which are introduced, probably fewer than a third of the inequalities of “Analytic Inequalities” are proved in the book. In fact the style of this book is perhaps closer to that usually associated with the “Ergebnisse” series (in which BB appears) than to that of the “Grundlehren” series. Unfortunately, one of the consequences of this style (especially in the last third of Part 2, and in Part 3) is the tendency for trivial and deep results to appear almost side-by-side, with little warning to the reader as to the category in which a given result falls.

The book is divided into three parts, and for a brief summary of these parts, we quote from the author’s Preface: “In the first part – “Introduction” – an approach to inequalities is given, while the main attention is devoted to the Section on Convex Functions. The second and probably main part – “General Inequalities” – consists of 27 sections, each of which is dedicated to a class of inequalities of importance in Analysis ... Finally, the third part – “Particular Inequalities” – is aimed at providing a collection of various inequalities, more or less closely interconnected, some of which are of considerable theoretical interest. They are classified in a certain manner, although we must admit that this has not been done perfectly. Part 3 is, in fact, a collection of over 450 special inequalities, and with a few exceptions we were able to add bibliographical references for each one. Owing to lack of space, only a few inequalities are supplied with a complete proof.”

Rather than trying to give a detailed list of the extremely large number of results in this volume, the reviewer believes it would be more useful to comment on some of those results which are not to be found in HLP or in BB. In Part 1 some extensions and generalizations of the concept of convex functions due to I. E. Ovcharenko, M. A. Krasnosel’skiĭ and Ya. B. Rutiskiĭ, and Á. Császár are given, without proof. In Part 2, recent refinements and extensions of such fundamental inequalities as those of Bernoulli, Cauchy, Chebyshev, Hölder, and Minkowski are given.

Section 2.10 deals with inequalities due to J. Aczel, T. Popoviciu, S. Kurepa, and R. Bellman of which the following one of Popoviciu is typical: If \(p\ge 1\), and \(a= (a_1,\ldots,a_n)\) and \(b=(b_1,\ldots,b_n)\) are sequences of nonnegative numbers such that \(A = a_1^p - a_2^p - \ldots a_n^p > 0\) or \(B =b_1^p - b_2^p - \ldots b_n^p > 0\), then \(AB \le (a_1b_1 - a_2b_2 - \ldots - a_nb_n)^p\).

In Section 2.11 certain “inverse” inequalities of Cauchy’s and Hölder’s inequalities due to J. B. Diaz and F. T. Metcalf, and Z. Nehari are given. Although the interesting inequality of G. Grüss dates back to 1935, it appears to have been overlooked in both HLP and BB. This inequality states that if \(f\) and \(g\) are Riemann integrable over \([a,b]\), with \(\varphi \le f(x) \le \Phi\) and \(\gamma\le g(x)\le \Gamma\) for all \(x\), then \[ \left\vert \frac1{b-a} \int_a^b fg\,dx - \frac1{(b-a)^2} \int_a^b f\,dx \int_a^b g \,dx \right\vert \le \frac14 (\Phi -\varphi) (\Gamma - \gamma), \] and the constant \(1/4\) is best possible. This result is proved in Section 2.13, and several generalizations and extensions are stated without proof.

Section 2.18 is devoted to a presentation (without proof) of certain inequalities involving the modulus of the sums of powers of complex numbers, due to P. Turán, introduced in his book “Eine neue Methode in der Analysis and deren Anwendungen.” Budapest: Akadémiai Kiadó (1953; Zbl 0052.04601), and to more recent extensions and improvements of these results by F. V. Atkinson and others.

In Section 2.19 the authors present a new elementary method for proving inequalities involving integrals of functions and their derivatives, due to D. C. Benson [J. Math. Anal. Appl. 17, 292–308 (1967; Zbl 0146.07404)].

A similar presentation of the method of “recurrent inequalities” of R. Redheffer [Prot. Lond. Math. Soc. (3) 17, 683–699 (1967; Zbl 0156.05804)] is given in Section 2.20; this powerful method applies only to discrete inequalities.

One of the longest sections in the book deals with two integral inequalities involving derivatives – Wirtinger’s inequality and Opial’s inequality – and with the numerous generalizations and extensions of these inequalities. (Incidentally, the authors point out that the name Wirtinger should probably not be attached to this inequality because it had been considered somewhat earlier by a number of other authors.)

In Section 2.25 several inequalities for vector norms are given, one of the most interesting and useful being the following reduction theorem of F. W. Levi [Arch. Math. 2, 24–26 (1949/50; Zbl 0034.29604)]: Let \(k\) and \(P_{ij}\) \((i = 1, \ldots,n; j = 1,\ldots, r)\) be real constants. Suppose that for all real numbers \(x_1,\ldots, x_r\) we have \[ \sum_{i=1}^n k_i\, \vert p_{i1}x_1 + \ldots + p_{ir}x_r\vert \ge 0. \] Then \[ \sum_{i=1}^n k_i\, \Vert p_{i1}a_1 + \ldots + p_{ir}a_r\Vert \ge 0 \] holds for arbitrary vectors \(a_1, \ldots,a_r\in E^*\) and any positive integer \(m\).

Part 3 is the longest part of the book (200 pages, compared to 26 and 160 pages for Parts 1 and 2). It contains nine sections bearing the following titles: Inequalities involving functions of discrete variables (11 pp.); Inequalities involving algebraic functions (19 pp.); Inequalities involving polynomials (18 pp.); Inequalities involving trigonometric functions (12 pp.); Inequalities involving trigonometric polynomials (19 pp.); Inequalities involving exponential, logarithmic and gamma functions (23 pp.); Integral inequalities (21 pp.); Inequalities in the complex domain (26 pp.); Miscellaneous inequalities (50 pp.). A surprisingly large number of the inequalities listed in Part 3 are drawn from the problems (and other) sections of such journals as the Mathematical Gazette, Mathematics Magazine, Mathematics Student, Wisk. Opgaven, and especially from the American Mathematical Monthly.

A further indication of the encyclopedic character of this volume can be inferred from the fact that there are over 750 names cited in the Name Index. (An interesting feature of this index is that the dates of birth, or the dates of birth and death, are given for practically all entries.)

Bibliographic references are given at the end of each section or subsection throughout the book. A much higher proportion of references to Slavic sources is given here than in either HLP or BB.

In the reviewer’s opinion this book is a valuable contribution to the field, and should be on the bookshelf of mathematicians, engineers, statisticians, physicists, and all who use inequalities in their work. By and large, it is probably safe to say that if an inequality – or a reference to a kind of inequality – which one requires can not be found in “Analytic Inequalities”, then either it is one of the special kind specifically excluded from this volume as noted above, or it can likely not be found anywhere.

Reviewer: Paul R. Beesack (Ottawa)

### MSC:

26Dxx | Inequalities in real analysis |

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

30A10 | Inequalities in the complex plane |

### Citations:

Zbl 0047.05302; Zbl 0126.28002; Zbl 0052.04601; Zbl 0146.07404; Zbl 0156.05804; Zbl 0034.29604
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\textit{D. S. Mitrinović}, Analytic inequalities. In cooperation with P. M. Vasić. Berlin: Springer-Verlag (1970; Zbl 0199.38101)

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### Digital Library of Mathematical Functions:

Jordan’s Inequality ‣ §4.18 Inequalities ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions§4.32 Inequalities ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions

§4.5(ii) Exponentials ‣ §4.5 Inequalities ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions

§4.5(i) Logarithms ‣ §4.5 Inequalities ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions

§7.8 Inequalities ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals