1.8 Some Inequalities in Common Use
71
a k and b k are expressed as x k and y k respectively, there is
n
k=1
|x k ||y k |
n
i=1
|x i |
p
1
p
n
i=1
|y i |
q
1
q
≤ 1
or
n
k=1
|x k ||y k | ≤
n
k=1
|x k |
p
1
p
n
k=1
|y k |
q
1
q
When the two series on the right side of the above expression converge, let n → ∞,
the inequality (1.8.4) can be obtained. Quod erat demonstrandum.
The Hölder inequality can also be written in integral form
t 1
t 0
|x(t)y(t)|dt ≤
t 1
t 0
|x(t)|
p dt
1
p
t 1
t 0
|y(t)|
q dt
1
q
(1.8.5)
Proof For the Young inequality (1.8.3), let
a =
|x(t)|
t 1
t 0
|x(t)|
p dt
1
p
, b =
|y(t)|
t 1
t 0
|y(t)|
q dt
1
q
Imitate the above method of proof, the inequality (1.8.5) can be derived. Quod
erat demonstrandum.
Suppose that R is a real number field, arbitrary real numbers x k , y k ∈ R, k =
1, 2, . . ., then there is
∞
k=1
x k y k ≤
∞
k=1
x
2
k
1
2
∞
k=1
y
2
k
1
2
(1.8.6)
When k > n, x k = y k = 0, the form of finite sum can be obtained. When the two
series on the right side of the above expression converge, it can be derivative that the
series on the left side converges. Equation (1.8.6) is called the Cauchy inequality.
Proof Let
a k =
x k
n
i=1
x
2
i
1
2
, b k =
y k
n
i=1
y
2
i
1
2