Convex functions, partial orderings, and statistical applications
type: Item
instance of: scholarly article
title: Convex functions, partial orderings, and statistical applications
zbMATH Open document ID: 0749.26004
author: Frank Proschan
author: Y. L. Tong
author: J. Pečarić
published in: Mathematics in Science and Engineering
publication date: 1992-09-18T00:00:00Z
review text: 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: ``\dots If \(F_ n\) is a quasi-arithmetic mean (for definition see Bullen,\dots) then \dots''. 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…
reviewed by: János Aczél
Mathematics Subject Classification ID: 26A51
Mathematics Subject Classification ID: 26-01
Mathematics Subject Classification ID: 26-02
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 26D20
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 53764
zbMATH Keywords: means
zbMATH Keywords: partial orderings
zbMATH Keywords: convex functions
zbMATH Keywords: inequalities
zbMATH Keywords: majorization
zbMATH Keywords: random variables
zbMATH Keywords: moments
zbMATH Keywords: density functions
zbMATH Keywords: distributions
zbMATH Keywords: arrangement ordering
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 53764