On the invariance of the arithmetic mean with respect to generalized Bajraktarević means
The purpose of this paper is to investigate the following invariance equation involving two 2-variable generalized Bajraktarević means, i.e., we aim to solve the functional equation f - 1 ( p 1 ( x ) f ( x ) + p 2 ( y ) f ( y ) p 1 ( x ) + p 2 ( y ) ) + g - 1 ( q 1 ( x ) g ( x ) + q 2 ( y ) g ( y )...
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| Published in | Acta mathematica Hungarica Vol. 166; no. 2; pp. 594 - 613 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.04.2022
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Summary: | The purpose of this paper is to investigate the following invariance equation involving two 2-variable generalized Bajraktarević means, i.e., we aim to solve the functional equation
f
-
1
(
p
1
(
x
)
f
(
x
)
+
p
2
(
y
)
f
(
y
)
p
1
(
x
)
+
p
2
(
y
)
)
+
g
-
1
(
q
1
(
x
)
g
(
x
)
+
q
2
(
y
)
g
(
y
)
q
1
(
x
)
+
q
2
(
y
)
)
=
x
+
y
(
x
,
y
∈
I
)
,
where
I
is a nonempty open real interval and
f
,
g
:
I
→
R
are continuous, strictly monotone and
p
1
,
p
2
,
q
1
,
q
2
:
I
→
R
+
are unknown functions. The main result of the paper shows that, assuming four times continuous differentiability of
f
,
g
, twice continuous differentiability of
p
1
and
p
2
and assuming that
p
1
differs from
p
2
on a dense subset of
I
, a necessary and sufficient condition for the equality above is that the unknown functions are of the form
f
=
u
v
,
g
=
w
z
,
and
p
1
q
1
=
p
2
q
2
=
v
z
,
where
u
,
v
,
w
,
z
:
I
→
R
are arbitrary solutions of the second-order linear differential equation
F
′
′
=
γ
F
(
γ
∈
R
is arbitrarily fixed) such that
v
> 0 and
z
> 0 holds on
I
and
{
u
,
v
}
and
{
w
,
z
}
are linearly independent. |
|---|---|
| ISSN: | 0236-5294 1588-2632 |
| DOI: | 10.1007/s10474-022-01230-5 |