Advanced High-School Mathematics

154 CHAPTER 3 Inequalities

x

y

y=x

n.

.

.

.

Q=Q(b,b )

n

P=P(a,a )

n

(S (a,b),0)n

Equation 1: y=x²

12. Show that ifa, b >0, then
    (a) S− 1 (a,b) = HM(a,b);
    (b) S 1 / 2 (a,b) = GM(a,b);
    (c) S 2 (a,b) = AM(a,b).


13. Show that

Sn(a,b) =

(n−1)(an−bn)

n(an−^1 −bn−^1 )

, (a 6 =b).

14. Show that if 2 ≤ m ≤ n are integers, and if a, b > 0 are real
    numbers, thenSm(a,b)≤Sn(a,b). (Hint: this can be carried out
    in a way similar to the solution of Exercise 11.

(a) First note that

(m−1)(am−bm)

m(am−^1 −bm−^1 )

≤

(n−1)(an−bn)

n(an−^1 −bn−^1 )

if an only if

n(m−1)(am−bm)(an−^1 −bn−^1 )≤m(n−1)(an−bn)(am−^1 −bm−^1 ).

(b) Next, define the polynomial

P(a,b) =m(n−1)(an−bn)(am−^1 −bm−^1 )−n(m−1)(am−bm)(an−^1 −bn−^1 );