zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Convex functions and their applications. A contemporary approach. (English) Zbl 1100.26002
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 (2005).

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(·) satisfying inequalities of the form

f(M(x,y))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{|u|,|v|}) 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.

MSC:
26-02Research monographs (real functions)
26A51Convexity, generalizations (one real variable)
26D15Inequalities for sums, series and integrals of real functions
49J52Nonsmooth analysis (other weak concepts of optimality)
46A55Convex sets in topological linear spaces; Choquet theory