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An Introduction to Inequalities

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Titlu: An Introduction to Inequalities

Autori Xiong Bin, Iurie Boreico, Vo Quoc Ba Can, Vasile Cirtoaje, Mircea Lascu
Pagini 264

Disponibilitate: în stoc.

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This book gathers these methods and tricks into a compact guide. It starts from the very basics, but covers almost all elementary methods known today (aside from a few very advanced ones). At higher levels of competition, good contestants are pretty much expected to know every technique listed here - meaning that there do appear inequalities that fit every presented category. Conversely, the vast majority of inequalities that do appear may be assigned to one of the sections of the book.

That being said, the present work is not meant as a textbook, or a comprehensive training material. It is very compact and fast-paced. There are a few examples for every subject, but to fully assimilate said subject, one needs to look elsewhere for more practice problems. The reader who truly starts ”from scratch” may quickly find him/herself overwhelmed by the pace and the increasing difficulty of the content; may this entirely normal impression not deter the student from pushing through!

Rather, the book is meant as crash course introduction to all facets of the subject, and as a reference guide. The readerwill get acquainted with all themethods, but must practice each tool on his/her own, using other sources. The one truth about competitive math is that solving problems is the most important part of learning. This holds just as true for inequalities and fortunately there are a lot of them out there to be used for training purposes.


1 Introductory notions
1.1 The concept of inequality
1.2 The basic properties of an inequality
1.3 Basic inequalities
1.4 Squares are positive
1.5 Some special inequalities and identities for two numbers
1.6 Some special inequalities and identities for more numbers
2 Basic techniques for solving inequalities
2.1 Comparison Method
2.2 Pigeonhole principle
2.3 Majorization and minorization
2.4 The method of undetermined coefficients
2.5 Normalization Method
2.6 Homogenization Method
2.7 The quadratic trinomial
2.8 Using equality to prove inequality
2.9 The mathematical induction
3 Some classical results and their applications
3.1 The AM-GM inequality
3.2 Young’s Inequality
3.3 Cauchy-Bunyakovsky-Schwarz Inequality
3.4 The Rearrangement Inequality
4 Techniques from Algebra
4.1 Breaking the inequality
4.2 Separating the squares
4.3 Working backwards
4.4 The Dual Principle and other standard substitutions
4.5 Mixing variables
4.6 Homogenization and dehomogenization
4.7 The inequalities between the symmetric sums
4.8 The pqr technique
5 Techniques from Analysis
5.1 The principle of extremality and monotonicity
5.2 Limits in inequalities
5.3 Derivatives in Inequalities
5.4 Convexity
5.5 Jensen’s Inequality
5.6 The shrinkage principle and Karamata’s Inequality
5.7 Schur’s Inequality
5.8 The generalized Means
5.9 The tangent line technique and its extensions
6 Glossary
7 Bibliography

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