REAL ANALYSIS
INEQUALITIES
The Inequality of Arithmetic and Geometric Means
Given
x
1
≥
0
x
2
≥
0
. . . x
n
≥
0
A
=
x
1
+
x
2
+
. . .
+
x
n
n
G
= (
x
1
x
2
. . . x
n
)
1
n
Then
G < A
unless
x
1
=
x
2
=
. . .
=
x
n
, when
G
=
A
.
Proof
Suppose without loss of generality that
x
1
is a maximal
x
ν
and
x
2
is
a minimal
x
ν
.
If
x
1
=
x
2
the
r
’s are all equal and there is nothing to prove.
Suppose then that
x
1
> x
2
. We form a new set of numbers
x
11
x
21
. . . x
n
1
by writing
x
11
=
A x
21
=

A
+
x
1
+
x
2
x
r
1
=
x
r
r
= 3
, . . . , n.
Let
A
1
, G
1
be the A.M and G.M of the
x
r
1
’s.
A
1
=
A
since
x
11
+
x
21
=
x
1
+
x
2
.
However
x
11
x
2
x

x
1
x
2
=
A
(
x
1
+
x
2

A
)

x
1
x
2
=
(
x
1

A
)(
A

x
2
)
>
0
since
x
1
> A > x
2
.
Therefore
G
1
> G
If the
x
ν
1
are not all equal we can again take a largest
x
α
1
and a smallest
x
β
1
and replace them by
A, x
α
1
+
x
α
2

A
.
A
2
=
A
1
G
2
> G
1
After at most
n

1 steps all the
x
’s are equal.
G < G
1
< . . . < G
K
=
A
k
=
A
therefore
G < A
.
Cauchy’s Inequalities
Given
x
1
. . . x
n
y
1
. . . y
n
real.
Then
1
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n
X
r
=1
x
r
y
r
≤
ˆ
n
X
r
=1
a
2
r
!
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 Spring '98
 pfitz
 Inequalities

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