Unknown

7.3. Inequalities 223

9. (a) Prove by induction on n that, if zi 2 -2 and xi all have the
   same sign, then

When does equality hold?

(b) Deduce from (a) the Bernoulli inequality

(1 + x)” > 1 + nz

for nonzero x >_ -2 and n a positive integer exceeding 1.

10. Newton’s inequalities. Let n be a positive integer and suppose that
    x1,x2,**, x,, are positive real numbers. For P = 1,2,... , n, define
    s, = cz122. “X,, the rth elementary symmetric function
    CXlXZ 2,
    Uf =
    n ’
    ( >

the average of the products of r numbers

v, = uy. r

Observe that vi is the ordinary arithmetic mean and vn the ordinary

geometric mean of the xi, so that v, 2 vr. This can be generalized

to the chain of inequalities

(a) Let f(t) = H{(t -xi) : i = 1,2,... ,n}. Show that

f(t) = i!” + 2(-l)’ ( : ) d+--‘.

r=l

(b) Use Rolle’s Theorem to argue that, for k = 1,2,... , n - 1, the

Cc)

(4

kth derivative fck)(t) is a polynomial of degree n - k with n - k

real positive zeros counting multiplicity.

Verify that f(n-2)(t) = (n!/2)(t2 - 2ult + ~2) and deduce from

the discriminant condition that 212 5 $.

Note that

n

( >

G-2^1 1

2

-= -+...+-

ull 21x2 +a-1%

n

( >

-= %-I

(^1) %l $+..
.+;
and apply (c) to l/Xi to obtain u,-2u, 5 u:-1.