Advanced High-School Mathematics

150 CHAPTER 3 Inequalities

then we see that AM(x 1 ,...,xn)≤QM(x 1 ,...,xn) is true without the
assumption that allxi are positive.

GM(x 1 ,...,xn)≤AM(x 1 ,...,xn): LetC= n

√

x 1 x 2 ···xn. If allxi=C,
then

√nx 1 x 2 ···xn=C= x 1 +···+xn

n

,

and we’re done in this case. Therefore, we may assume that at least
one of thexis is less thanC and that one is greater thanC. Without
loss of generality, assume that x 1 > C and that C > x 2. Therefore,

(x 1 −C)(C−x 2 ) > 0 and sox 1 x 2 < C(x 1 +x 2 )−C^2 ⇒
x 1 +x 2
C

>

Çx
1
C

åÇx

2

C

å

+ 1. From this, we conclude

x 1 +x 2 +···+xn

C

>

(x 1 x 2 )/C+x 3 +···+xn

C

+ 1

≥ (n−1)n−^1

√

(x 1 x 2 ···xn)/Cn+ 1 (using induction)

= (n−1) + 1 =n.

That is to say, in this case we have
x 1 +x 2 +···+xn
n

> C= n

√

x 1 x 2 ···xn,

concluding the proof that GM≤ AM. (For a much easier proof, see
Exercise 2 on page 160.)

HM(x 1 ,...,xn)≤GM(x 1 ,...,xn): From the above we get

1

x 1 +

1

x 2 +···+

1

x 3

n

≥ n

Ã

1

x 1

1

x 2

···

1

xn

;

take reciprocals of both sides and infer that HM≤GM.

A generalization of AM ≤ QM is embodied in the very classical
Cauchy-Schwarz inequality. We state this as a theorem.

Theorem 1. (Cauchy-Schwarz Inequality)Given
x 1 , x 2 , ..., xn, y 1 , y 2 , ..., yn∈R, one has

(x 1 y 1 +x 2 y 2 +···+xnyn)^2 ≤(x^21 +x^22 +···+x^2 n)(y 12 +y 22 +···+yn^2 ).