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China National Olympiad
1989 China National Olympiad
6
6
Part of
1989 China National Olympiad
Problems
(1)
China Mathematical Olympiad 1989 problem6
Source: China Mathematical Olympiad 1989 problem6
10/29/2013
Find all functions
f
:
(
1
,
+
∞
)
→
(
1
,
+
∞
)
f:(1,+\infty) \rightarrow (1,+\infty)
f
:
(
1
,
+
∞
)
→
(
1
,
+
∞
)
that satisfy the following condition: for arbitrary
x
,
y
>
1
x,y>1
x
,
y
>
1
and
u
,
v
>
0
u,v>0
u
,
v
>
0
, inequality
f
(
x
u
y
v
)
≤
f
(
x
)
1
4
u
f
(
y
)
1
4
v
f(x^uy^v)\le f(x)^{\dfrac{1}{4u}}f(y)^{\dfrac{1}{4v}}
f
(
x
u
y
v
)
≤
f
(
x
)
4
u
1
f
(
y
)
4
v
1
holds.
function
inequalities
algebra unsolved
algebra