## 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.

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Received: October 5, 1998; revised version: March 15, 1999.

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Losonczi, L. Equality of two variable weighted means: reduction to differential equations.
*Aequ. math.* **58**, 223–241 (1999). https://doi.org/10.1007/s000100050110

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DOI: https://doi.org/10.1007/s000100050110