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2019 Korea USCM
6
6
Part of
2019 Korea USCM
Problems
(1)
Holder like inequality
Source: 2019 South Korea USMC P6
8/13/2020
A function
f
:
[
0
,
∞
)
→
[
0
,
∞
)
f:[0,\infty)\to[0,\infty)
f
:
[
0
,
∞
)
→
[
0
,
∞
)
is integrable and \int_0^\infty f(x)^2 dx<\infty, \int_0^\infty xf(x) dx <\infty Prove the following inequality.
(
∫
0
∞
f
(
x
)
d
x
)
3
≤
8
(
∫
0
∞
f
(
x
)
2
d
x
)
(
∫
0
∞
x
f
(
x
)
d
x
)
\left(\int_0^\infty f(x) dx \right)^3 \leq 8\left(\int_0^\infty f(x)^2 dx \right) \left(\int_0^\infty xf(x) dx \right)
(
∫
0
∞
f
(
x
)
d
x
)
3
≤
8
(
∫
0
∞
f
(
x
)
2
d
x
)
(
∫
0
∞
x
f
(
x
)
d
x
)
real analysis
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