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Miklós Schweitzer
2005 Miklós Schweitzer
8
functional equation
functional equation
Source: Miklos Schweitzer 2005 q8
August 27, 2021
functional equation
real analysis
algebra
Problem Statement
Determine all continuous, strictly monotone functions
ϕ
:
R
+
→
R
\phi : \mathbb{R}^+\to\mathbb{R}
ϕ
:
R
+
→
R
such that
F
(
x
,
y
)
=
ϕ
−
1
(
x
ϕ
(
x
)
+
y
ϕ
(
y
)
x
+
y
)
+
ϕ
−
1
(
y
ϕ
(
x
)
+
x
ϕ
(
y
)
x
+
y
)
F(x,y)=\phi^{-1} \left(\frac{x\phi(x)+y\phi(y)}{x+y}\right) + \phi^{-1} \left(\frac{y\phi(x)+x\phi(y)}{x+y}\right)
F
(
x
,
y
)
=
ϕ
−
1
(
x
+
y
x
ϕ
(
x
)
+
y
ϕ
(
y
)
)
+
ϕ
−
1
(
x
+
y
y
ϕ
(
x
)
+
x
ϕ
(
y
)
)
is homogeneous of degree 1, ie
F
(
t
x
,
t
y
)
=
t
F
(
x
,
y
)
,
∀
x
,
y
,
t
∈
R
+
F(tx,ty)=tF(x,y) , \forall x,y,t\in\mathbb{R}^+
F
(
t
x
,
t
y
)
=
tF
(
x
,
y
)
,
∀
x
,
y
,
t
∈
R
+
F(x,y)=F(y,x) and F(x,x)=2x
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