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Vojtěch Jarník IMC
2003 VJIMC
Problem 4
double integral inequality in two functions
double integral inequality in two functions
Source: VJIMC 2003 2.4
July 13, 2021
calculus
integration
inequalities
function
Problem Statement
Let
f
,
g
:
[
0
,
1
]
→
(
0
,
+
∞
)
f,g:[0,1]\to(0,+\infty)
f
,
g
:
[
0
,
1
]
→
(
0
,
+
∞
)
be two continuous functions such that
f
f
f
and
g
f
\frac gf
f
g
are increasing. Prove that
∫
0
1
∫
0
x
f
(
t
)
d
t
∫
0
x
g
(
t
)
d
t
d
x
≤
2
∫
0
1
f
(
t
)
g
(
t
)
d
t
.
\int^1_0\frac{\int^x_0f(t)\text dt}{\int^x_0g(t)\text dt}\text dx\le2\int^1_0\frac{f(t)}{g(t)}\text dt.
∫
0
1
∫
0
x
g
(
t
)
d
t
∫
0
x
f
(
t
)
d
t
d
x
≤
2
∫
0
1
g
(
t
)
f
(
t
)
d
t
.
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