Equality of two variable weighted means: reduction to differential equations

Summary.

Let $\Phi, \Psi$ be strictly monotonic continuous functions, F,G be positive functions on an interval I and let $n \in {\Bbb N} \setminus \{1\}$. The functional equation¶¶$\Phi^{-1}\,\left({\sum\limits_{i=1}^{n}\Phi(x_{i})F(x_{i})\over\sum\limits_{i=1}^{n} F(x_{i}}\right) \Psi^{-1}\,\left({\sum\limits_{i=1}^{n}\Psi(x_{i})G(x_{i})\over\sum\limits_{i=1}^{n} G(x_{i})}\right)\,\,(x_{1},\ldots,x_{n} \in I)$¶was solved by Bajraktarević [3] for a fixed $n\ge 3$. Assuming that the functions involved are twice differentiable he proved that the above functional equation holds if and only if¶¶$\Psi(x) = {a\Phi(x)\,+\,b\over c\Phi(x)\,+\,d},\qquad G(x) = kF(x)(c\Phi(x) + d)$¶where a,b,c,d,k are arbitrary constants with $k(c^2+d^2)(ad-bc)\ne 0$. Supposing the functional equation for all $n = 2,3,\dots$ Aczél and Daróczy [2] obtained the same result without differentiability conditions.¶The case of fixed n = 2 is, as in many similar problems, much more difficult and allows considerably more solutions. Here we assume only that the same functional equation is satisfied for n = 2 and solve it under the supposition that the functions involved are six times differentiable. Our main tool is the deduction of a sixth order differential equation for the function $\varphi = \Phi\circ\Psi^{-1}$. We get 32 new families of solutions.