Inequalities involving functions and their integrals and derivatives. (English) Zbl 0744.26011

The first monograph on inequalities in the literature is the book “Inequalities” (1934; Zbl 0010.10703); 2nd ed. (1952; Zbl 0047.05302); reprint (1988; Zbl 0634.26008) by G. H. Hardy, J. E. Littlewood and G. Pólya. Since that time the subject of this book has grown about 800%. Several books on such inequalities have appeared. For example, the books “Inequalities” (1961; Zbl 0097.26502); 4th print (1983, Zbl 0513.26003) by E. F. Beckenbach and R. Bellman, “Analytic inequalities” (1970; Zbl 0199.38101) by D. S. Mitrinović, “Differential und Integralungleichungen” (1964; Zbl 0119.12205); English translation (1970; Zbl 0252.35005) by W. Walter, “Differential and integral inequalities” (Vols. I and II) (Vol. I: 1969; Zbl 0177.12403) by V. Lakshmikantham and S. Leela, “Theorems on inequalities” (Russian) Ashkhabad: Ylym (1980) by Ya. D. Mamedov, S. Ashirov and S. Atdaev, and “Stability of motion: The method of integral inequalities” (Russian) Kiev: Naukova Dumka (1989; Zbl 0751.70014) by A. A. Martynyuk, V. Lakshmikantham and S. Leela. However, these volumes mentioned did not attempt to give comprehensive account of differential or integral inequalities.
The book under review gives an up to date, comprehensive survey of inequalities involving a relationship between a function and its derivatives or/and integrals. The authors write : “On the one hand, we have undertaken the task of doing an exhaustive search of the literature on differential and integral inequalities. On the other hand in such an undertaken, the number of papers devoted to differential and integral inequalities is so large that space does not allow us to quote all the results or prove most of those that we do quote.”…“In writing a systematic account we take this view that there are certain fundamental inequalities, perhaps very simple to state, that are the heart of the matter.”
The authors have attempted to classify each result as being closely related to a “well-known classical result”. The review consists of 18 chapters. The chapter headings are as follows: Landau-Kolmogorov and related inequalities; An inequality ascribed to Wirtinger and related results; Opial’s inequality; Hardy’s, Carleman’s and related inequalities; Hilbert’s and related inequalities; Inequalities of Lyapunov and of de la Vallée Poussin; Zmorovich’s and related inequalities; Inequalities involving kernels; Convolution, rearrangement and related inequalities; Inequalities of Chaplygin type; Inequalities of Gronwall type of a single variable; Gronwall inequalities in higher dimension; Gronwall inequalities in other spaces; Integral inequalities involving functions with bounded derivatives; Inequalities of Bernstein-Mordell type; Methods of proofs for integral inequalities; Particular inequalities.
This book is not for beginners – rather, it is more of a research monograph written for people who already have some understanding of the field, because less than a quarter of the quoted results are proved in the book and applications are given only for Gronwall type and Chaplygin type inequalities. The omitted proofs draw heavily on the specialized literature and an inexperienced reader will need much patience and effort in going back to the original papers if he (or she ) wants to fill the details of all the proofs. There are some typographical errors in the book. However, I found no serious errors.
This book is a valuable addition to the literature and it will be an essential part of every mathematical library. Congratulations and thanks are due to the authors.


26D10 Inequalities involving derivatives and differential and integral operators
26-02 Research exposition (monographs, survey articles) pertaining to real functions
26D15 Inequalities for sums, series and integrals
34A40 Differential inequalities involving functions of a single real variable