MathDB
Problems
Contests
International Contests
Hungary-Israel Binational
2000 Hungary-Israel Binational
3
kl positive integers
kl positive integers
Source: 11-th Hungary-Israel Binational Mathematical Competition 2000
April 22, 2007
inequalities
induction
inequalities unsolved
Problem Statement
Let
k
k
k
and
l
l
l
be two given positive integers and
a
i
j
(
1
≤
i
≤
k
,
1
≤
j
≤
l
)
a_{ij}(1 \leq i \leq k, 1 \leq j \leq l)
a
ij
(
1
≤
i
≤
k
,
1
≤
j
≤
l
)
be
k
l
kl
k
l
positive integers. Show that if
q
≥
p
>
0
q \geq p > 0
q
≥
p
>
0
, then
(
∑
j
=
1
l
(
∑
i
=
1
k
a
i
j
p
)
q
/
p
)
1
/
q
≤
(
∑
i
=
1
k
(
∑
j
=
1
l
a
i
j
q
)
p
/
q
)
1
/
p
.
(\sum_{j=1}^{l}(\sum_{i=1}^{k}a_{ij}^{p})^{q/p})^{1/q}\leq (\sum_{i=1}^{k}(\sum_{j=1}^{l}a_{ij}^{q})^{p/q})^{1/p}.
(
j
=
1
∑
l
(
i
=
1
∑
k
a
ij
p
)
q
/
p
)
1/
q
≤
(
i
=
1
∑
k
(
j
=
1
∑
l
a
ij
q
)
p
/
q
)
1/
p
.
Back to Problems
View on AoPS