A short remark on the arithmetic mean of involutions (Q2568004)

An involution is a function whose second iterate is the identity. Let \(i_1,i_2: \mathbb{R}\to\mathbb{R}\) be continuous involutions with 0 as fixed point and \(\phi\) the involution defined by \(x\mapsto c| x|^a\) for \(x<0, x\mapsto -(x/c)^{1/a}\) for \(x\geq 0.\) The author offers the result that \((i_1+i_2)/2=\phi\) on \(\mathbb{R}\) implies \(i_1=i_2=\phi.\)