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TitleAn Introduction to Inequalities
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Total Pages142
Table of Contents
                            An Introduction to Inequalities
	Front Matter
		Note to the Reader
		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
		1.5 Additional Inequality Relationships
		1.6 Products Involving Negative Numbers
		1.7 "Positive" and "Negative" Numbers
		Exercises
	2. Tools
		2.1 Introduction
		2.2 Transitivity
		2.3 Addition
		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
		Answers to Exercises
			Chapter 1
			Chapter 2
			Chapter 3
			Chapter 4
			Chapter 5
			Chapter 6
		Index
                        

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