The associativity equation. Synthesis of ratio judgements. The quasiarithmetic means. The Jensen equations. A conditional linear-affine equation. A characterization of root-mean-powers and of the geometric mean

Abstract

While we have often dealt with functional equations for multiplace functions in the present work, the translation equations (6.6) and (6.21) are different: They are composite equations: the unknown function appears again inside the unknown function (on the right hand side). Here we will solve another important composite equation

$$F\left[{F\left({x,y}\right),z}\right]=F\left[{x,F\left({y,z}\right)}\right]\left({x,y,z\in I,F:I^2\to I}\right)$$

(1)