MathDB
Problems
Contests
Undergraduate contests
Vojtěch Jarník IMC
1999 VJIMC
Problem 4
integral inequality in [0,1]^n
integral inequality in [0,1]^n
Source: VJIMC 1999 2.4
July 29, 2021
calculus
integration
Problem Statement
Let
u
1
,
u
2
,
…
,
u
n
∈
C
(
[
0
,
1
]
n
)
u_1,u_2,\ldots,u_n\in C([0,1]^n)
u
1
,
u
2
,
…
,
u
n
∈
C
([
0
,
1
]
n
)
be nonnegative and continuous functions, and let
u
j
u_j
u
j
do not depend on the
j
j
j
-th variable for
j
=
1
,
…
,
n
j=1,\ldots,n
j
=
1
,
…
,
n
. Show that
(
∫
[
0
,
1
]
n
∏
j
=
1
n
u
j
)
n
−
1
≤
∏
j
=
1
n
∫
[
0
,
1
]
n
u
j
n
−
1
.
\left(\int_{[0,1]^n}\prod_{j=1}^nu_j\right)^{n-1}\le\prod_{j=1}^n\int_{[0,1]^n}u_j^{n-1}.
(
∫
[
0
,
1
]
n
j
=
1
∏
n
u
j
)
n
−
1
≤
j
=
1
∏
n
∫
[
0
,
1
]
n
u
j
n
−
1
.
Back to Problems
View on AoPS