Unknown

224 7. Approximations and Inequalities

(e) The results of (c) and (d) can be generalized to u,-ru,+r <

uz for r = 2,3,..., n - 1. This is established by an induction

argument on the number of the xi. The result holds for n = 3.

Suppose it holds when the number of the ti does not exceed

n - 1. From (a), obtain that

f’(t) = n

1

t”-’ + “e(-1)’ ( n f ’ ) u.t”-‘-‘].

t-=1

(f) Suppose the zeros of f’(t) are ~1,... , ~~-1. For 1 < r 5 n - 1,

let

& = CYlY2 “‘Yrr,

so that the Z, are to the yi what the u, are to the xi. Use the

argument of (a) to obtain

f@) = n

1

p-1 +ng(-l)‘. ( n ; l ) %.t”-‘-‘1.

?-El

11.

12.

Deduce that u, = zr for 1 5 P 2 n - 1. Use the induction

hypothesis to obtain u,-iu,+i < u,” for 2 5 r 5 n - 2.

(g) Use (c), (d), (e), (f) to obtain for 1 5 P 5 n - 1,

u2(ulu3)2(u2u4)3... (UT-1U,+#^5 u’4u;u; *.. up,

and, hence

u,+l P 5 u:+l, i.e. v,+i 5 v,.

Suppose that a, b, c are nonnegative reals for which (1 + a)(1 + b)

(1 + c) = 8. We can apply the result of Exercise 10 to show that

ubc 5 1. Simply show that the left side of the given condition is not

less than (1 + u)~ where u3 = abc.

Suppose that a, b, c are positive real numbers. Show that

a+;+:<

a” + bs + 8

a3b3c3 ’

This can be established by repeated use of the AGM inequality. To

get started, write the right side as the sum of three terms of the form

a5/b3c3 and apply the AGM inequality to the sum of each pair.

Exploration

E.61. Can the polynomial x5 + y5 + ,z5 + u5 + v5 - 5xyzuv be manipu-
lated into a form in which it is clearly seen to be nonnegative for x, y, t,
u, v nonnegative? Consider the analogous question for other numbers of
variables.