# Books for inequality proofs

I was wondering: what books for proving inequalities are used in universities when studying mathematics (undergraduate)?

I know there are lots of books, but I would like to know which ones are specifically used in math studies.

If you mean what would be suitable reference/resource books for undergraduates to know about, three books I know about that have withstood the test of time are:

Inequalities by Hardy/Littlewood/Pólya

Introduction to Inequalities by Beckenbach/Bellman

Geometric Inequalities by Kazarinoff

Hardy's book is probably too advanced in many places for most undergraduates, but it's such a classic that it has to be included. The other two are less advanced and would be suitable for lower level undergraduates and even good high school students. If you have access to a university library, you can also go to where the Hardy book is shelved and look in the same vicinity for other books on inequalities. For this it doesn't matter whether Hardy's book is checked out, since it's purpose here is simply to position you at the appropriate location in the library stacks.

• Thank you very much for your answer! – Jan Sep 5 '14 at 15:51

I'm afraid that this question, which is a reasonable question, does not have a reasonable, concise answer. Inequalities come in a few different forms. These might be classified as

$1$. Fundamental Inequalities $2$. Inequalities that appear in basic studies, such as calculus $3$. Inequalities that appear in advanced studies,such as functional analysis $4$. Olympiad-style inequalities

Finding $x$ that satisfies inequalities like $(x+1)^2 + x > 0$, or more generally $p(x) > 0$ for some polynomial $p(x)$, is largely too basic to be covered in its own book. This is also true for inequalities like $\lvert x + 2 \rvert - 2 > 0$. Instead, these are often associated with "college algebra" or "precalculus" textbooks (note: college algebra $\neq$ modern algebra or abstract algebra).

A new batch of inequalities are possible once you hit calculus, as calculus provides techniques for proving inequalities in great generality. Inequalities here might include $1 + nx < (1 + x)^n < e^{nx}$. This highlights a certain problem, which is that there are many interesting or useful inequalities, but they rely on other material for proof. So they appear most naturally in other courses.

This happens again in higher mathematics, but I feel is sufficiently different to warrant its own classification. Functional analysis or PDE (or many other topics) use and inform many inequalities. As an analytic number theorist, much of what I do can be summarized as "proving inequalities on L-functions and Dirichlet Series".

Finally, there is the class of very interesting and often very fun olympiad style inequalities. These can often be proved from first principles, and I think they're very enjoyable. But they also don't appear in the "standard undergraduate curriculum", whatever that might mean. For these, I highly recommend the books "Problem Solving through Problems" and "Problem Solving Strategies" (which are both great general olympiad preparation books, and common supplements to Putnam-centric courses), and the "Cauchy-Schwarz Master Class" (which is also a book, but entirely inequality focused).

I hope this helps.

• Thank you for the answer! What is the difference between normal inequalities and Olympiad inequalities? – Jan Sep 5 '14 at 15:33
• Olympiad inequalities, like many olympiad questions, often rely on some fundamental trick, while requiring almost no theory to prove. – davidlowryduda Sep 5 '14 at 21:37