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TitleAn Introduction to Inequalities
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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
                        
Document Text Contents
Page 1

AN INTRODUCTION
TO INEQUALITIES
EDWIN BECKENBACH
RICHARD BELLMAN

The L. W. Singer Company
New Mathematical Library

Page 71

62 A N I N T R O D U C T I O N T O I N E Q U A L I T I E S
3. Show that the arithmetic mean of two positive numbers is less than or equal

to their root-mean-square:
a + b < ja2 + b2

2 - 2
Under what circumstance does the sign of equality hold? How does the
root-mean-square compare with the geometric mean and with the har­
monic mean?

4. Let ABDC be a trapezoid with AB = a , CD = b (see Fig. 4.4). Let 0
be the point of intersection of its diagonals. Show that

Figure 4.4 Geometric Illustration of 2ab < . !(if- < a + b < �
a + b

- v uu - 2 - V -y--
(a) The anthmeuc mean (a + b)/2 of a and b is represented by the line
segment GH parallel to the bases and halfway between them.
(b) The geometric mean yaTi is represented by the line segment KL
parallel to the bases and situated so that trapezoids ABLK and KLDC are
similar.
(c) The harmonic mean is represented by the line segment EF parallel to
the bases and passing through 0.
(d) The root-mean-square is represented by the line segment MN parallel
to the bases and dividing the trapezoid ABDC into two trapezoids of equal
area.

4.4 The Cauchy Inequality

(a) The Two-dimensional Version: (a2 + b2)(c2 + d2) 2:: (ac + bd)2.
Let us now introduce a new theme. As in a musical composition, this
theme will intertwine with the original theme to produce further and
more beautiful results.

We begin with the observation that the inequality

a2 + b2 2:: 2ab .
on which all the proofs in the preceding sections of this chapter were
based [see Sec. 4.2(b)], is a simple consequence of the identity

a2 - 2ab + b2 = (a - b)2 ,

Page 72

T H E C L A S S I C A L I N E Q U A L I T I E S

which is valid for all real a, b, not merely for nonnegative a, b.
Consider now the product

(4.35) (a2 + b2)(c2 + d2) .
We see, upon multiplying out, that this product yields

a2c2 + b2d2 + a2d2 + b2c2 ,

63

which is identically what we obtain from expanding the expression

(4.36) (ac + bd)2 + (be - ad)2 .
Hence we have

(4.37) (a2 + b2) (c2 + d2) = (ac + bd)2 + (be - ad)2 .
Since the square (be - ad )2 is nonnegative, from (4.37) we obtain

(4.38) (a2 + b2) (c2 + d2) 2:: (ac + bd)2 , for all real a, b, c, d,
a very pretty inequality of great importance throughout much of
analysis and mathematical physics. It is called the Cauchy inequality,
or, more precisely, the two-dimensional version of the Cauchy
inequality.t

Furthermore, we see from (4.37) that the sign of equality holds in
( 4.38) if and only if

(4.39) be - ad = 0 .

In this case, we say that the two pairs (a, b) and (c, d ) are proportional
to each other; if c =¥= 0 and d =¥= 0 , the condition ( 4.39) can be
written as

(b) Geometric Interpretation. On first seeing the identity of the
expressions (4.35) and (4.36), the reader should quite naturally and
legitimately wonder how in the world anyone would ever stumble
upon this result. It strikes him as having been "pulled out of a hat,"
a piece of mathematical sleight of hand.

It is a tenet of a mathematician's faith that there are no accidents
in mathematics. Every result of any significance has an explanation

t A generahzation of this inequality to expressions occurnng in integral calculus
was discovered independently by the mathematicians Buniakowski and Schwarz. The
name "Cauchy-Schwarz inequality" is sometimes applied to the inequality in the text,
but more particularly to its generalization.

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