A Gauss type functional equation (Q5939668)

Let \(p: \mathbb{R}_+\to \mathbb{R}_+\) be strictly monotonic and define \(f(a,b;p)= \int_0^{2\pi} p(a\cos^2\theta+ b\sin^2\theta) d\theta/(2\pi)\) and \(M_p(a,b)= p^{-1} (f(a,b;p))\) for \(a,b> 0\). NEWLINENEWLINENEWLINEThe authors find necessary conditions on \(p\) for \(f\) to be a solution of the functional equation \(F(M(a,b), N(a,b))= F(a,b)\), given the means \(M\) and \(N\). A mean is a function \(M: \mathbb{R}_+\times \mathbb{R}_+\to \mathbb{R}_+\) such that \(\min(a,b)\leq M(a,b)\leq \max(a,b)\) for all \(a,b> 0\). NEWLINENEWLINENEWLINEThey give necessary conditions on \(p\) for \(M_p= N\) to hold, given the mean \(N\).