Multivariable generalizations of bivariate means via invariance

For a given p-variable mean \(M :I^p \rightarrow I\) $M:I^p\to I$ (I is a subinterval of \({\mathbb {R}}\) $\mathbb{R}$ ), 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 \((p+1)\) $(p+1)$ -variable \(\beta \) $\beta$ -invariant extension as the unique solution \(K :I^{p+1} \rightarrow I\) $K:I^{p+1}\to I$ of the functional equation \(\begin{aligned}&K\big (M(x_2,\dots ,x_{p+1}),M(x_1,x_3,\dots ,x_{p+1}),\dots ,M(x_1,\dots ,x_p)\big )\\&\quad =K(x_1,\dots ,x_{p+1}), \text { for all }x_1,\dots ,x_{p+1} \in I \end{aligned}\) $$\begin{array}{cc}& K(M(x_2,\cdots ,x_{p+1}),M(x_1,x_3,\cdots ,x_{p+1}),\cdots ,M(x_1,\cdots ,x_p))\hfill \\ \hfill & =K(x_1,\cdots ,x_{p+1}),\text{for all}{x}_1,\cdots ,x_{p+1}\in I\hfill \end{array}$$ 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.