aequationes mathematicae

, Volume 58, Issue 3, pp 223–241

Equality of two variable weighted means: reduction to differential equations

  • L. Losonczi

DOI: 10.1007/s000100050110

Cite this article as:
Losonczi, L. Aequ. math. (1999) 58: 223. doi:10.1007/s000100050110


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.

Keywords. Mean value, functional equation.

Copyright information

© Birkhäuser Verlag, Basel, 1999

Authors and Affiliations

  • L. Losonczi
    • 1
  1. 1.Department of Mathematics and Computer Science, Kuwait University, P.O.Box 5969, Safat, 13060, Kuwait, e-mail: