Advanced High-School Mathematics

160 CHAPTER 3 Inequalities

Indeed, from the above, we have by settingr=

q

p

>1 that

∑n

i=1

xip/n = AM(xp 1 ,xp 2 ,...,xpn)

≤ rM(xp 1 ,xp 2 ,...,xpn)

=

Ñn

∑

i=1

(xpi)q/p/n

ép/q

=

Ñ n

∑

i=1

xqi/n

ép/q

.

Taking thep-th roots of both sides yields what we were after, viz.,

pM(x 1 ,x 2 ,...,xn) =

Ñ n

∑

i=1

xip/n

é 1 /p

≤

Ñn

∑

i=1

xqi/n

é 1 /q

= qM(x 1 ,x 2 ,...,xn).

Exercises.

1. Show how Young’s inequality proves that GM(x 1 ,x 2 )≤AM(x 1 ,x 2 ),
   wherex 1 ,x 2 ≥0.


2. Use Jensen’s inequality and the fact that the graph ofy= lnxis
   concave down to obtain a simple proof that

AM(x 1 ,x 2 ,...,xn)≥GM(x 1 ,x 2 ,...,xn),

wherex 1 ,x 2 ,...,xn≥0.

3. Use Jensen’s inequality to prove that given interior anglesA, B,
   andC of a triangle then

sinA+ sinB+ sinC≤ 3

√

2 / 2.

Conclude that for a triangle 4 ABC inscribed in a circle of radius

R, the maximum perimeter occurs for an equilateral triangle. (See

Exercise 2 on page 34.)

4. Given 4 ABCwith areaKand side lengthsa, b,andc, show that

ab+ac+bc≥ 4

√

3 K.

Under what circumstances does equality obtain? (Hint: note that

6 K =absinC+acsinB+bcsinA; use Cauchy-Schwarz together

with Exercise 3, above.)