Author: Pham Van Thuan, Le Vi

This book is intended as a useful resource for high school and college students
who are training for national or international mathematical competitions.
But anybody who is interested in elementary mathematical inequalities may find
this book useful. This problem solving book is divided into six chapters,
containing more than fifty topics of interest to mathematical olympiad contestants
and coaches, demonstrating ideas and strategies in solving inequalities.
The reader will find in the book clever applications of well known results as
well as powerful original methods, each is explained and illustrated by
carefully selected problems.
Chapter 1. Inequalities between means. This chapter starts with the fundamental fact $x^2\ge 0$, upon which many
interesting inequalities are derived. It also serves to emphasize that powerful
results can be obtained by little means.

Chapter 2. Cauchy-Schwars. The classical result of Cauchy-Schwarz is revisited with examples illustrating
sophisticated, at time surprising, ways to apply the inequality. Other
classical inequalities such as those of Chebyshev and H\”older are also
discussed and shown how they might cooperate with Cauchy-Schwarz inequality.

Chapter 3. Convexity. This chapter utilises calculus in solving inequalities. Based on simpe
properties of linear and convex functions, systematic methods are derived
to tackle some advanced problems. Also discussed is tangent line method,
which gives a geometric interpretation of bounds.

Chapter 4. Homogenous inequalities. Homogeneous inequalities constitutes a large class of inequality problems.
This chapter discusses various approaches to solving this class of inequalities, including the techniques of homogenization, normalisation, the application of Rolle’s theorem to reduce the number of variables, the use of limits and partitions, quadratic estimations, and establishing new bounds through isolated fudging. Especially in focus are powerful techniques to solve inequalities by the change of variables $p=a+b+c$, $q=ab+bc+ca$ and $=abc$ and by
transforming them to one of the following forms.

\$latex x(a-b)(a-c)+y(b-c)(b-a)+z(c-a)(c-b)\ge 0\$,

$x(a-b)^2+y(b-c)^2+z(c-a)^2 \ge 0$,

$M(a-b)^2 + N(c-a)(c-b) \ge 0$.
All three, four variable symmetric polynomial inequalities can be solved using ideas in this chapter.

Chapter 5. The method of Mixing Variables. The method of mixing variables has been used in various forms for decades – an
example is G. Polya’s delightful proof of the AM-GM inequalities. This chapter
examines this idea in depth with extension in different directions.
The first three sections explain why mixing variables work, give hints to
find approriate variables to mix by taking equality cases into consideration.
The most important results in this chapter are two theorems which facilitates
solutions for a large class of multi-variable inequalities.
Chapter 6. Further Topics and problems with solutions. The chapter starts with miscellaneous indenpendent topics touching upon various aspects of solving inequalities. The discussion includes
the interplay between trogonometric and algebraic substitution,
absolute values, inequalities with special equality cases and
inequalities with ordered sequences.

For pdf file: sample-ineqs.pdf

1. Pirquliyev Rovsen says:

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