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Vojtěch Jarník IMC
2014 VJIMC
Problem 4
double integral inequality
double integral inequality
Source: VJIMC 2014 2.4
May 21, 2021
inequalities
integration
calculus
Problem Statement
Let
0
<
a
<
b
0<a<b
0
<
a
<
b
and let
f
:
[
a
,
b
]
→
R
f:[a,b]\to\mathbb R
f
:
[
a
,
b
]
→
R
be a continuous function with
∫
a
b
f
(
t
)
d
t
=
0
\int^b_af(t)dt=0
∫
a
b
f
(
t
)
d
t
=
0
. Show that
∫
a
b
∫
a
b
f
(
x
)
f
(
y
)
ln
(
x
+
y
)
d
x
d
y
≤
0.
\int^b_a\int^b_af(x)f(y)\ln(x+y)dxdy\le0.
∫
a
b
∫
a
b
f
(
x
)
f
(
y
)
ln
(
x
+
y
)
d
x
d
y
≤
0.
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