##### Document Text Contents

Page 1

AN INTRODUCTION

TO INEQUALITIES

EDWIN BECKENBACH

RICHARD BELLMAN

The L. W. Singer Company

New Mathematical Library

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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 ,

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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.

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.