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48
Functional equation with α, β
Functional equation with α, β
Source:
September 2, 2010
function
algebra
functional equation
IMO Shortlist
IMO Longlist
Problem Statement
Find all the functions
f
:
R
+
→
R
f : \mathbb R^+ \to \mathbb R
f
:
R
+
→
R
satisfying the identity
f
(
x
)
f
(
y
)
=
y
α
f
(
x
2
)
+
x
β
f
(
y
2
)
∀
x
,
y
∈
R
+
f(x)f(y)=y^{\alpha}f\left(\frac x2 \right) + x^{\beta} f\left(\frac y2 \right) \qquad \forall x,y \in \mathbb R^+
f
(
x
)
f
(
y
)
=
y
α
f
(
2
x
)
+
x
β
f
(
2
y
)
∀
x
,
y
∈
R
+
Where
α
,
β
\alpha,\beta
α
,
β
are given real numbers.
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