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Vojtěch Jarník IMC
2007 VJIMC
Problem 4
weak convexity, prove inequality
weak convexity, prove inequality
Source: VJIMC 2007 1.4
June 24, 2021
inequalities
function
Problem Statement
Let
f
:
[
0
,
1
]
→
[
0
,
∞
)
f:[0,1]\to[0,\infty)
f
:
[
0
,
1
]
→
[
0
,
∞
)
be an arbitrary function satisfying
f
(
x
)
+
f
(
y
)
2
≤
f
(
x
+
y
2
)
+
1
\frac{f(x)+f(y)}2\le f\left(\frac{x+y}2\right)+1
2
f
(
x
)
+
f
(
y
)
≤
f
(
2
x
+
y
)
+
1
for all pairs
x
,
y
∈
[
0
,
1
]
x,y\in[0,1]
x
,
y
∈
[
0
,
1
]
. Prove that for all
0
≤
u
<
v
<
w
≤
1
0\le u<v<w\le1
0
≤
u
<
v
<
w
≤
1
,
w
−
v
w
−
u
f
(
u
)
+
v
−
u
w
−
u
f
(
w
)
≤
f
(
v
)
+
2.
\frac{w-v}{w-u}f(u)+\frac{v-u}{w-u}f(w)\le f(v)+2.
w
−
u
w
−
v
f
(
u
)
+
w
−
u
v
−
u
f
(
w
)
≤
f
(
v
)
+
2.
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