2.1 Identities and Inequalities 39

2.1.5 The Arithmetic Mean–Geometric Mean Inequality.............

Jensen’s inequality, which will be discussed in the section about convex functions, states
that iffis a real-valued concave function, then

f(λ 1 x 1 +λ 2 x 2 +···+λnxn)≥λ 1 f(x 1 )+λ 2 f(x 2 )+···+λnf(xn),

for anyx 1 ,x 2 ,...,xnin the domain offand for any positive weightsλ 1 ,λ 2 ,...,λn
withλ 1 +λ 2 +···+λn=1. Moreover, if the function is nowhere linear (that is, if it is
strictly concave) and the numbersλ 1 ,λ 2 ,...,λnare nonzero, then equality holds if and
only ifx 1 =x 2 = ··· =xn.
Applying this to the concave functionf(x)=lnx, the positive numbersx 1 ,x 2 ,...,
xn, and the weightsλ 1 =λ 2 = ··· =λn=^1 n, we obtain

ln

x 1 +x 2 +···+xn

n

≥

lnx 1 +lnx 2 +···+lnxn

n

.

Exponentiation yields the following inequality.

The arithmetic mean–geometric mean inequality.Letx 1 ,x 2 ,...,xnbe nonnegative
real numbers. Then

x 1 +x 2 +···+xn

n

≥ n

√

x 1 x 2 ···xn,

with equality if and only if all numbers are equal.

We will call this inequality AM–GM for short. We give it an alternative proof using
derivatives, a proof by induction onn. Forn=2 the inequality is equivalent to the
obvious(

√

a 1 −

√

a 2 )^2 ≥0. Next, assume that the inequality holds for anyn−1 positive
numbers, meaning that

x 1 +x 2 +···+xn− 1

n− 1

≥ n−^1

√

x 1 x 2 ···xn− 1 ,

with equality only whenx 1 =x 2 = ··· =xn− 1. To show that the same is true forn
numbers, consider the functionf:( 0 ,∞)→R,

f(x)=

x 1 +x 2 +···+xn− 1 +x

n

−n

√

x 1 x 2 ···xn− 1 x.

To find the minimum of this function we need the critical points. The derivative offis

f′(x)=

1

n

−

√nx 1 x 2 ···xn− 1

n

x

(^1) n− 1

x
n^1 −^1
n

(

x^1 −

(^1) n
−n

√

x 1 x 2 ···xn− 1

)

.