7.3. Inequalities 223
- (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.
- 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.