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Convex functions, partial orderings, and statistical applications. (English) Zbl 0749.26004

Mathematics in Science and Engineering. 187. Boston, MA etc.: Academic Press. xiii, 467 p. (1992).
There are many books on inequalities and on convex functions but then there are so many inequalities around — more or less connected to convex functions — that no book of reasonable size can contain all. The present book takes a pretty good stab at it — as long as the inequalities are satisfied by convex functions in general and, except in Chapters 13-17, not by particular ones (thus showing a similar trait to some complex function theory courses where the only complex function a student sees is “\(f\)”). So, for instance, when following the subject index the reader looks up “Minkowski’s inequality” he or she may find something different from what could be expected.
Speaking about subject indices, this reviewer having stood on both sides of the fence, can appreciate that the typical author finds it tiresome to put them together. But the typical reader, in particular of such encyclopedical works (not meant, we hope, to be read cover to cover in linear order) relies heavily on them. So, for example, “monotonic \(n\)- tuple”, “isotonic functional” might be sorely missed (only “inequalities” are under I) as well as properties \(A1,A2,\dots,L1,L2,K_ 1,K_ 2\) in the index or in the list of notations.
The book is not self-contained: Not only are results mentioned only by reference, but so are even definitions, for instance in Theorem 8.3: “…If \(F_ n\) is a quasi-arithmetic mean (for definition see Bullen,…) then …”. Since there is no proof, the reader can understand this theorem only if she or he looks it up elsewhere. This is quite strange because either a direct definition or specialization from the “generalized mean” defined on p. 107 would take only a couple of lines. — Also, not all definitions are marked as such. — In Remark 15.22, counterexamples would be appreciated which show why the two conjectures there formulated are false.
Chapters 1-10 are solidly filled wall to wall with inequalities; even the word “application” refers there to other inequalities. Chapter 11 contains “geometric” inequalities, Chapter 12 majorization and Schur convexity, while Chapters 13-17 give applications to probability and statistics (random variables, their moments, density functions, distributions, arrangement ordering).
All over there are many interesting historical remarks, including corrections to conventional wisdom about who has priority to what definition or result. {L. Fejér is on the receiving end of one of these remarks. Another quotation from Fejér may be of interest in this context: “The history of mathematics serves to prove that nobody discovered anything: there was always somebody who knew it before.”} Some of these remarks also correct published statements and point out simpler proofs (usually by the authors). All in all, a very useful book, but there is room for improvement.

MSC:

26A51 Convexity of real functions in one variable, generalizations
26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
26-02 Research exposition (monographs, survey articles) pertaining to real functions
26D15 Inequalities for sums, series and integrals
26D10 Inequalities involving derivatives and differential and integral operators
26D20 Other analytical inequalities
60E15 Inequalities; stochastic orderings
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