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Inequalities

Theorems, Techniques and Selected Problems

  • Book
  • © 2012

Overview

Authors:
  1. Zdravko Cvetkovski

    1. of the Republic of Macedonia, Faculty of Informatics, European University, Skopje, Macedonia

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  • Contains 485 problems on Algebra-Inequalities

  • Detailed solutions are given

  • Theoretical background and techniques for proving inequalities are provided

  • Ideal for participants of national and international mathematical contests

  • Includes supplementary material: sn.pub/extras

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Table of contents (21 chapters)

  1. Front Matter

    Pages I-X
    Download chapter PDF

  2. Basic (Elementary) Inequalities and Their Application

    • Zdravko Cvetkovski
    Pages 1-7

  3. Inequalities Between Means (with Two and Three Variables)

    • Zdravko Cvetkovski
    Pages 9-18

  4. Geometric (Triangle) Inequalities

    • Zdravko Cvetkovski
    Pages 19-25

  5. Bernoulli’s Inequality, the Cauchy–Schwarz Inequality, Chebishev’s Inequality, Surányi’s Inequality

    • Zdravko Cvetkovski
    Pages 27-47

  6. Inequalities Between Means (General Case)

    • Zdravko Cvetkovski
    Pages 49-60

  7. The Rearrangement Inequality

    • Zdravko Cvetkovski
    Pages 61-67

  8. Convexity, Jensen’s Inequality

    • Zdravko Cvetkovski
    Pages 69-77

  9. Trigonometric Substitutions and Their Application for Proving Algebraic Inequalities

    • Zdravko Cvetkovski
    Pages 79-94

  10. Hölder’s Inequality, Minkowski’s Inequality and Their Variants

    • Zdravko Cvetkovski
    Pages 95-105

  11. Generalizations of the Cauchy–Schwarz Inequality, Chebishev’s Inequality and the Mean Inequalities

    • Zdravko Cvetkovski
    Pages 107-116

  12. Newton’s Inequality, Maclaurin’s Inequality

    • Zdravko Cvetkovski
    Pages 117-119

  13. Schur’s Inequality, Muirhead’s Inequality and Karamata’s Inequality

    • Zdravko Cvetkovski
    Pages 121-132

  14. Two Theorems from Differential Calculus, and Their Applications for Proving Inequalities

    • Zdravko Cvetkovski
    Pages 133-136

  15. One Method of Proving Symmetric Inequalities with Three Variables

    • Zdravko Cvetkovski
    Pages 137-145

  16. Method for Proving Symmetric Inequalities with Three Variables Defined on the Set of Real Numbers

    • Zdravko Cvetkovski
    Pages 147-153

  17. Abstract Concreteness Method (ABC Method)

    • Zdravko Cvetkovski
    Pages 155-160

  18. Sum of Squares (SOS Method)

    • Zdravko Cvetkovski
    Pages 161-167

  19. Strong Mixing Variables Method (SMV Theorem)

    • Zdravko Cvetkovski
    Pages 169-175

  20. Method of Lagrange Multipliers

    • Zdravko Cvetkovski
    Pages 177-182
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Keywords

About this book

This work is about inequalities which play an important role in mathematical Olympiads. It contains 175 solved problems in the form of exercises and, in addition, 310 solved problems. The book also covers the theoretical background of the most important theorems and techniques required for solving inequalities. It is written for all middle and high-school students, as well as for graduate and undergraduate students. School teachers and trainers for mathematical competitions will also gain benefit from this book.

Reviews

From the reviews:

“The book is aimed at the more advanced students about to enter university and is particularly useful to candidates, and their trainers, for mathematical competitions such as the Olympiads. The themes and methods are introduced in 19 chapters of the book, which include exercises and their solutions.” (Peter Shiu, The Mathematical Gazette, Vol. 98 (541), March, 2014)

“This volume collects problems of various degrees of difficulty in the field of elementary inequalities. This book is intended as a valuable source for the training of high-school students in view of mathematical Olympiads. School teachers will also gain benefit from this book. … It would be a particularly valuable resource for those who participate in mathematics competitions at the high school or college level. … this book is a ‘must have’ for a university’s library, and I recommend it highly to its ‘ideal audience’.” (Teodora-Liliana Radulescu, Zentralblatt MATH, Vol. 1233, 2012)

Authors and Affiliations

  • of the Republic of Macedonia, Faculty of Informatics, European University, Skopje, Macedonia

    Zdravko Cvetkovski

About the author

Dipl. Math. Zdravko Cvetkovski, European University-Skopje, R. Macedonia, Informatics Department.

Bibliographic Information

  • Book Title: Inequalities

  • Book Subtitle: Theorems, Techniques and Selected Problems

  • Authors: Zdravko Cvetkovski

  • DOI: https://doi.org/10.1007/978-3-642-23792-8

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2012

  • Softcover ISBN: 978-3-642-23791-1Published: 06 January 2012

  • eBook ISBN: 978-3-642-23792-8Published: 06 January 2012

  • Edition Number: 1

  • Number of Pages: X, 444

  • Topics: Algebra, Popular Science, general, Mathematics Education

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