Title An Introduction to Inequalities 1.6 MB 142
```                            An Introduction to Inequalities
Front Matter
Contents
Preface
1. Fundamentals
1.1 The "Greater-than" Relationship
1.2 The Sets of Positive Numbers, Negative Numbers, and Zero
1.3 The Basic Inequality Axioms
1.4 Reformulation of Axiom I
1.6 Products Involving Negative Numbers
1.7 "Positive" and "Negative" Numbers
Exercises
2. Tools
2.1 Introduction
2.2 Transitivity
2.4 Multiplication by a Number
2.5 Subtraction
Exercises
2.6 Multiplication
2.7 Division
Exercises
2.8 Powers and Roots
Exercises
3. Absolute Value
3.1 Introduction
3.2 Definition
3.3 Special Symbols
Exercises
3.4 Graphical Considerations
Exercises
3.5 The "Sign" Function
Exercises
3.6 Graphs of Inequalities
Exercises
3.7 Algebraic Characterization
Exercises
3.8 The "Triangle" Inequality
Exercises
4. The Classical Inequalities
4.2 The Inequality of the Arithmetic and Geometric Means
A. Mathematical Experimentation
Exercises
B. Proof of the Arithmetic-mean-Geometric-mean Inequality for Two Numbers
C. A Geometric Proof
D. A Geometric Generalization
Exercises
E. The Arithmetic-mean-Geometric-mean Inequality for Three Numbers
F. The Arithmetic-mean-Geometric-mean Inequality for n Numbers
4.3 Generalizations of the Arithmetic-mean-Geometric-mean Inequality
Exercises
4.4 The Cauchy Inequality
A. The Two-dimensional version
B. Geometric Interpretation
C. Three-Dimensional Version of the Cauchy Inequality
D. The Cauchy-Lagrange Identity and the n-Dimensional Version of the Cauchy inequality
E. An Alternative Proof of the Three-Dimensional Version
4.5 The Holder Inequality
4.6 The Triangle Inequality
4.7 The Minkowski Inequality
4.8 Absolute Values and the Classical Inequalities
4.9 Symmetric Means
4.10 The Arithmetic-Geometric Mean of Gauss
5. Maximization and Minimization Problems
5.1 Introduction
5.2 The Problem of Dido
5.3 Simplified Version of Dido's Problem
5.4 The Reverse Problem
5.5 The Path of a Ray of Light
5.6 Simplified Three-dimensional Version of Dido's Problem
Exercises
5.7 Triangles of Maximum Area for a Fixed Perimeter
Exercises
5.8 The Wealthy Football Player
5.9 Tangents
5.10 Tangents (Concluded)
Exercises
6. Properties of Distance
6.1 Euclidean Distance
6.2 City-Block Distance
6.3 Some Other Non-Euclidean Distances
6.4 Unit Discs
Exercises
6.5 Algebra and Geometry
Back Matter
Symbols