On the mutual noncompatibility of homogeneous analytic non-power means
Homogeneous symmetric means \(\mu\) and \(\tilde \mu\) defined on \(\mathbb{R}^ n_ +\) and \(\mathbb{R}_ +^{n+1}\), respectively, are called compatible if the value of \(\tilde \mu\) remains unchanged upon replacing \(n\) of its arguments by their \(\mu\)-mean. It is proved that, if \(\mu\) and \(\overline\mu\) are compatible continuous homogeneous means in \(\mathbb{R}^ n_ +\) and \(\mathbb{R}_ +^{n+1}\) respectively \((n \geq 2)\) and either \(\mu\) is analytic or \(\tilde\mu\) is analytic with strictly positive partial derivatives, then there exists \(r \in \mathbb{R}\) such that \(\mu=M_ n^{[r]}\), \(\tilde\mu=M^{[r]}_{n+1}\), where \(M_ n^{[r]}(x_ 1,\dots,x_ n)={1 \over n} (\sum^ n_{i=1}x^ r_ i)^{1/r}\) if \(r \neq 0\) and \(M^ 0_ n(x_ 1,\dots,x_ n)=(\prod^ n_{i=1} x_ i)^{1/n}\). This is proved by fixing all but one argument in both \(\mu\) and \(\tilde\mu\) which leads to the functional equation \(x+g(y+f(x))=y+g(x+f(y))\), \(x,y \in \mathbb{R}\). This equation is solved in the class of complex power series and local analytic functions at 0.
analytic solutions of functional equation
scientific article; zbMATH DE number 202392