Multivariable generalizations of bivariate means via invariance

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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.