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Functional inequality (BxMO 2023, Problem 1)
Functional inequality (BxMO 2023, Problem 1)
Source: BxMO 2023, Problem 1
May 6, 2023
BxMO
algebra
Problem Statement
Find all functions
f
:
R
→
R
f\colon\mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that
(
x
−
y
)
(
f
(
x
)
+
f
(
y
)
)
⩽
f
(
x
2
−
y
2
)
(x-y)\bigl(f(x)+f(y)\bigr)\leqslant f\bigl(x^2-y^2\bigr)
(
x
−
y
)
(
f
(
x
)
+
f
(
y
)
)
⩽
f
(
x
2
−
y
2
)
for all
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
.
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