The author describes the function of this work as follows: “[T]he object of this book is to provide an easy way for researchers to locate an inequality either by name or by subject (p. xi)”. The second edition includes a separate name index and a subject index, both of which were absent from the first. The work itself is arranged, in a dictionary fashion, alphabetically by inequality name. This access is supplemented with a large number of cross-references throughout the work that compensates reasonably for inconsistent naming practices of mathematical inequalities. Each entry includes the inequalities, with commentary and references, including url’s where available. The commentary often provides cross-references, synonyms to the inequality names among other comments. This work excludes the area of geometric inequalities and inequalities from number theory. The second edition reviewed here is significantly expanded – it is approximately one-third larger than the first edition published in 1998.
Researchers and librarians interested in the subject of mathematical inequalities may wish to consult a worthwhile bibliographic essay by Fink (2006) that covers several important books in this area. Bullen, in the preface, outlines many books and authors of importance in this field.
Dictionary of Inequalities can be purchased in print form as here reviewed or as an e-book through a variety of platforms, including that of the publisher. The e-book version is available via several aggregator platforms and under a variety of licenses and prices. These aggregator platforms currently include EBL, Ebrary, Ebschost, Amazon Kindle and the new Proquest EBook Central.
When the first edition of this book was published, the internet, while in use by academics, was a tiny fraction of the size and usefulness that it is today. Acknowledging this significant difference in the environment for scholarly literature leads this reviewer to evaluate whether or not a dictionary like this will be currently useful for scholars. The author points out, in his introductory remarks about sources for this work, “of course there is Wikipedia and Google that between them seem to be able to answer almost any question” (p. xiv). Indeed, many of the inequalities found in this book can be retrieved by searching with Google, and often the specific inequalities can be found in Wikipedia or the Encyclopedia of Mathematics (www.encyclopediaofmath.org/index.php/Main_Page) or Wolfram MathWorld (http://mathworld.wolfram.com). This reviewer was unable to easily retrieve many of the inequalities found in this book via the latter sources. Additionally, searching Google, the general web search, ran into familiar problems with many irrelevant results, and results of later research were related to inequality and not a dictionary or encyclopedic type sources of information as found in the work under review. Additionally, searching in this manner produced results with wide variation in the quality of sources. The entries in the Encyclopedia of Mathematics and Wolfram MathWorld were structured in a similar fashion to this work’s entries and frequently contained more references and details than this book. However, this dictionary contains a significantly greater number of inequalities than either of these sources and also much more than Wikipedia, whose entries are of widely varying content.
This reviewer is left wondering, if the work had been published as an open education resource instead of the traditional book format, would the readily availability to students and mathematicians around the world have made for a much more lasting impact on the world of mathematics and could it have been more readily added to or updated in the future? Despite this consideration, the structure of this work and its coverage of inequalities make it useful for those scholars needing to regularly and quickly look up mathematical inequalities. It will be a worthwhile purchase for a research library where there are programs in mathematics and for mathematicians who work with inequalities.
