CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 23. New York, NY: Springer (ISBN 0-387-24300-3/hbk). xvi, 255 p. EUR 59.95/net; $ 69.95; £ 46.00 (2006).

The book is devoted to elementary theory of convex functions. Among the series of books related to convex functions, one can see the following pecularities: (1) the authors give either the theory of convex functions on the real axis or the theory of convex functions on normed linear spaces; (2) for convex functions on the real axis the authors present the theory of some generalizations of convex functions; (3) the main object in this book is connected with analysis of different inequalities related to convex functions and their modifications; (4) the authors do not touch some thin aspects of the theory, in particular, some fine theorems about the existence of the second differentials everywhere and some others. The book consists of Preface, List of symbols, Introduction, four chapters and four supplements, References (253 items) and Index. Chapter 1 “Convex functions on intervals” deals with the most elementary definitions and theorems of the theory: discrete and integral Jensen’s inequalities, conjugate convex functions and classical Young’s inequality, subdifferentials and integral representation of convex functions, Hermite-Hadamard inequalities, relations between convexity and majorization and so on. Chapter 2 “Comparative convexity on intervals” presents elements of the theory for functions $f(\cdot)$ satisfying inequalities of the form $$f(M(x,y)) \le N(f(x),f(y))$$ for a suitable pair of means $M$ and $N$. In particular, the authors’ considerations cover log-convex functions, multiplicatively convex functions and the class of $M_p$-convex functions. The authors’ approach in this chapter seems to be interesting; however, it is a pity that they are restricted with the case when $M$ and $N$ are means. So, the important class of quasi-convex functions ($M$ is the arithmetical mean, $N(u,v) = \max \ \{\vert u\vert ,\vert v\vert \}$) turns out their arguments and constructions. Chapter 3 “Convex functions on normed linear spaces” is central in the book. The chapter deals with general definitions of convex functions, their continuity, differentiability and twice differentiability (here one can find classical theorems by Rademacher and A. D. Aleksandrov, but E. Asplund), Prékopa-Leindler type inequalities, and many other interesting things. Chapter 4 “Choquet’s theory and beyond” gives the description of Steffensen-Popoviciu measures, Jensen-Steffensen and Steffensen’s inequality, and, at last, Choquet’s theorem about Hermite-Hadamard inequality in the metrizable case and Choquet-Bishop-de Leeuw theorem. It is necessary to add, that each chapter presents numerous exercises and consists of deep and interesting comments. Supplement A “Background on convex sets” deals with the Hahn-Banach extension theorem in analytical and geometrical forms and the Krein-Milman theorem. Supplement B “Elementary symmetric functions” gives an acoount of Newton’s and Newton-like inequalities for elementary symmetric functions. Supplement C “The variational approach of PDE” illustrates a number of problems in partial differential equations which can be solved by seeking a global minimum of a suitable convex functional. Supplement D “Horn’s conjecture” presents some convexity properties for eigenvalues of matrices. In general, the book will take a worthy place among books by L. Hörmander, M. A. Krasnosel’skij-Y. B. Rutickij, J. E. Pečarić-F. Proschan-Y. C. Tong, R. R. Phelps, A. W. Roberts-D. T. Varberg that are devoted, completely or partly, to convex functions. The book will be useful to all who are interested in convex functions and their applications.