Multivariable generalizations of bivariate means via invariance

For a given p-variable mean \(M :I^p \rightarrow I\) (I is a subinterval of \({\mathbb {R}}\)), 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 \((p+1)\)-variable \(\beta \)-invariant extension as the unique solution \(K :I^{p+1} \rightarrow I\) of the functional equation

$$\begin{aligned}&K\big (M(x_2,\dots ,x_{p+1}),M(x_1,x_3,\dots ,x_{p+1}),\dots ,M(x_1,\dots ,x_p)\big )\\&\quad =K(x_1,\dots ,x_{p+1}), \text { for all }x_1,\dots ,x_{p+1} \in I \end{aligned}$$

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.