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The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities

J. Michael Steele
Mathematical Association of America
Publication Date: 
Number of Pages: 
Problem Books
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Leo Livshits
, on

When I was a graduate student, my supervisor relayed to me the following aphorism: "research mathematicians are people with solutions who are looking for problems." J. Michael Steele's The Cauchy-Schwarz Master Class, a guided tour on discovery of proofs of inequalities, will surely convince its readers of the validity of this tidbit of wisdom.

It must be as difficult to write an engaging book on inequalities as on plumber's wrenches. Hardy, Littlewood and Polya's famous Inequalities is the case in point: an excellent reference, but hardly readable. Steele's book is so readable and lively, one almost forgets the underlying material, and therein lies Steele's ingenious approach: the book's real subject is the process of building mathematics, with the inequalities simply supplying the context.

As Steele moves the train of thought from the simplest form of Cauchy-Schwarz to Rademacher-Menchoff, the stations en route flash new vistas which are deftly tamed with proofs that echo methods previously used, ideas discussed, tricks presented. The arguments look deceptively simple, and this illusion is reinforced by various " may not need long to observe...", " ...we are not likely to need long to think of looking for...", "...after some exploration one does discover...", " we seem to be in luck..." and "...this observation almost begs us to ask...". While personally I do not care for this speciosity, I do appreciate its soothing affect on an insecure reader, not to mention its helping hand in keeping the presentation concise.

As "The Cauchy-Schwarz Master Class" spans about 150 years worth of inequalities (discounting such outliers as AM-GM) whose discovery often required flights of genius and deep intuition, Steele is compelled, and justifiably so, to include historical and biographical digressions introducing the players involved. Frequent and satisfying in the first part of the book, these sadly grow more sparse in later chapters. One also wonders why Schur or Riesz are deemed less deserving of a significant "blurb" than Pólya. Fortunately the chapter notes (relegated to the back of the book) mitigate the discrepancy, and if these were to actually follow the corresponding individual chapters, they could easily carry a greater burden of history, to the book's advantage.

The Ideal Audience

The Cauchy-Schwarz Master Class is focused on proofs, and so is not suitable to the uninitiated. Neither it is appropriate for those requiring applied motivation as justification for a task; here things are done the "Pure Math" way.

Very little beyond series, multivariable calculus and basic complex notation is required to follow the steps of the proofs, but non-trivial mathematical sophistication is necessary to appreciate the grand plans. The "Master Class" declaration unties the author's hands to bring in the necessary concepts without the burden of extensive prior development. It is a credit to J. Michael Steele that he does this in a most non-threatening way possible.

Most senior undergraduates or "Masters+" students would find this a rewarding read. Exercises at the end of each section provide an opportunity for increased challenge, while pursuing some of the concepts further can give the endeavor a greater breadth. The fact that the book includes a section of concise solutions to the exercises makes it an even more attractive educational tool.

The words "Master Class" in the title carry a heavy load. In the music world, master classes usually involve several proficient and self-motivated musicians, and are led by an accomplished performer. There is little doubt that J. Michael Steele is an outstanding expositor, and as such he is justified to assume that his text can serve as a competent leader for a reasonably proficient and motivated student/practitioner of mathematics (or for a small group thereof), undertaking an independent study.

Just as it would not do for two divas to preside over the same master class, it would be rather challenging to teach a (seminar) course with The Cauchy-Schwarz Master Class as a textbook; the flow of text is too tight to serve two masters.

The play-by-play

Each of the 14 chapters of the roughly 300 page book is comprised of several theorems stated as problems, each followed by a solution framed as a discussion. [Calling theorems "problems" is a bit peculiar, especially since the book is not trying to be a "problem book" à la those of Paul R. Halmos.] Each chapter ends with a number of exercises (11-12 on average), solutions to which are given at the back. Most of the exercises seem natural and informative.

Chapter abstracts given below are by the reviewer, whereas chapter titles are original.

  • Chapter 1: Starting with Cauchy. Finite-dimensional Cauchy-Schwarz inequality. Real inner product notation and abstraction. Cauchy-Schwarz in real inner product spaces.
  • Chapter 2: Cauchy's Second Inequality: The AM-GM Bound. Discrete Arithmetic Mean-Geometric Mean and Carleman's inequalities.
  • Chapter 3: Lagrange's Identity and Minkowski's Conjecture. Equality in Cauchy-Schwarz in Rn. Lagrange identity for the defect in Cauchy-Schwarz in Rn. Minkowski's conjecture on existence of nonnegative polynomials in two variables that cannot be written as a sum of squares of real polynomials.
  • Chapter 4: On Geometry and Sums of Squares. Triangle inequality in Rn. Definition of normed linear spaces. A bound on the product of two linear forms. Reverse of Cauchy-Schwarz for Lorenz product. Cauchy-Schwarz for complex inner products.
  • Chapter 5: Consequences of Order. The hunt for a converse to Cauchy-Schwarz in Rn. Chebyshev's order inequality and its connection to probability. Rearrangement inequality.
  • Chapter 6: Convexity-The Third Pillar. Jensen's inequlity and Hölder's defect formula for Jensen.
  • Chapter 7: Integral Intermezzo. Integral version of Jensen's inequality. Miscellaneous integral inequalities.
  • Chapter 8: The ladder of Power Means. Weighted lp means (a.k.a. "power means") in Rn. Geometric mean as a limit of weighted lp means. Monotonicity of lp means. Weighted Lp means (integral analogues of "power means"). Termwise strengthening of Carleman's inequality.
  • Chapter 9: Hölder's Inequality. Hölder's inequality and its stability. Minkowski's inequality. lp-lq duality in Rn. An illustration of l1-linterpolation.
  • Chapter 10: Hilbert's Inequality and Compensating Difficulties. Hilbert's inequality and its converse.
  • Chapter 11: Hardy's Inequality and The Flop. Integral and discrete Hardy's inequalities. Carleson's convexity inequality.
  • Chapter 12: Symmetric Sums. Inequalities of Newton and Maclauren. Muirhead's inequality in Rn.
  • Chapter 13: Majorization and Schur Convexity. Schur's analytic criteria for Schur convexity (a.k.a. monotonicity with respect to majorization). Equivalence of Muirhead's, majorization and stochastic matrix conditions in Rn. Schur's majorization inequality.
  • Chapter 14: Cancellation and Aggregation. Abel's inequality. Exponential sums. van der Corput inequality. Rademacher-Menchoff inequality for partial sums of orthonormal series.
  • Solutions to Exercises.
  • Chapter Notes.
  • References.
  • Index.


Overall, this book is a "must have" for a university's library, and I recommend it highly to its "ideal audience." Many other readers are also bound to discover a satisfying number of attractive and less than familiar results. I, for one, have reserved a prominent space for the book on my shelf, and feel quite certain that I will leaf through it time and time again.

Leo Livshits is Associate Professor of Mathematics at Colby College.

1. Starting with Cauchy; 2. The AM-GM Inequality; 3. Lagrange's Identity and Minkowski's Conjecture; 4. On Geometry and Sums of Squares; 5. Consequences of Order; 6. Convexity — The Third Pillar; 7. Integral Intermezzo; 8. The Ladder of Power Means; 9. Hölder's Inequality; 10. Hilbert's Inequality and Compensating Diffculties; 11. Hardy's Inequality and the Flop; 12. Symmetric Sums; 13. Majorization and Schur Convexity; 14. Cancellation and Aggregation; Solutions to the Exercises; Chapter Notes; References