Abstract
AbstractFor a given p-variable mean $$M :I^p \rightarrow I$$
M
:
I
p
→
I
(I is a subinterval of $${\mathbb {R}}$$
R
), following (Horwitz in J Math Anal Appl 270(2):499–518, 2002) and (Lawson and Lim in Colloq Math 113(2):191–221, 2008), we can define (under certain assumptions) its $$(p+1)$$
(
p
+
1
)
-variable $$\beta $$
β
-invariant extension as the unique solution $$K :I^{p+1} \rightarrow I$$
K
:
I
p
+
1
→
I
of the functional equation $$\begin{aligned}&K\big (M(x_2,\dots ,x_{p+1}),M(x_1,x_3,\dots ,x_{p+1}),\dots ,M(x_1,\dots ,x_p)\big )\\&\quad =K(x_1,\dots ,x_{p+1}), \text { for all }x_1,\dots ,x_{p+1} \in I \end{aligned}$$
K
(
M
(
x
2
,
⋯
,
x
p
+
1
)
,
M
(
x
1
,
x
3
,
⋯
,
x
p
+
1
)
,
⋯
,
M
(
x
1
,
⋯
,
x
p
)
)
=
K
(
x
1
,
⋯
,
x
p
+
1
)
,
for all
x
1
,
⋯
,
x
p
+
1
∈
I
in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions.
Publisher
Springer Science and Business Media LLC
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