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Analytic Inequalities

Nicholas D. Kazarinoff
Dover Publications
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This is a concise introduction to the subject of inequalities aimed at a post-calculus undergraduate level. It makes heavy use of college algebra and assumes familiarity with calculus, especially the mean-value theorem, integrals, and infinite series. The qualifier “analytic” in the title is not completely accurate, as the book covers all kinds of inequalities, but it is slanted toward inequalities used in mathematical analysis.

The book includes a number of mathematical applications of inequalities, particularly in approximations. It includes a proof of the Weierstrass Approximation Theorem using Bernstein polynomials, a moderate amount on the isoperimetric theorem, and various error estimates for sequences, series, and products. On the inequality side of things, it has thorough coverage of the arithmetic mean–geometric mean inequality, with many applications. Although it proves the Cauchy-Schwarz, Hölder, and Minkowski inequalities, it doesn’t do much with them. The most conspicuous omission is Jensen’s inequality for convex functions, that appears only briefly in the exercises.

The book includes many problems scattered through the text, with a long list of miscellaneous problems at the end. This last section covers all areas of inequalities and mixes very easy and very difficult problems without distinction; the collection seems to be what was left over when the author ran out of room for exposition. Unfortunately none of the problems have any hints or solutions given.

This book is still up-to-date and is available at a bargain Dover price, but it suffers by comparison with more recent books. The present work is an unaltered reprint of the 1961 work published by Holt, Rinehart and Winston. When it came out, it was the only book-length treatment of inequalities except for Hardy & Littlewood & Pólya’s Inequalities, and that book was full of difficult notations and intricate reasoning and was (and still is) tough going for anyone. Today a better choice than the present book would be Steele’s The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. It is pitched at the same level or slightly higher, has broader coverage, and includes many new results obtained after Kazarinoff’s book was published. Another interesting book that focuses on mathematical applications is Niven’s Maxima and Minima Without Calculus. Niven uses classical inequalities, especially the arithmetic mean–geometric mean inequality, to solve most problems.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

  • Preface
  1. Fundamentals
    1. The algebra of inequalities
    2. Conditional inequalities
  2. Two Ancient Theorems
    1. Geometric and arithmetic means
    2. An application
    3. The Isoperimetric Theorem
  3. Inequalities and Calculus
    1. The number e
    2. Examples from the calculus
    3. Approximation by polynomials
  4. Three Modern Theorems
    1. Power means
    2. The Cauchy, Bunyakovskii, Hölder, and Minkowski Inequalities
  • Miscellaneous problems