For a given $p$-variable mean $M \colon I^p \to I$ ($I$ is a subinterval of $\mathbb{ R}$), following (Horwitz, 2002) and (Lawson and Lim, 2008), we can define (under certain assumption) its $(p+1)$-variable $\beta$-invariant extension as the unique solution $K \colon I^{ p+1} \to I$ of the functional equation \begin{ align*} 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)\\ &=K(x_1,\dots,x_{ p+1}), \text{ for all }x_1,\dots,x_{ p+1} \in I \end{ align*} 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.
Categories: math.DS