On a functional equation related to power means

In the paper On the mutual noncompatibility of homogeneous analytic non-power means (Aequationes Math. 45 (1993)) M. E. Kuczma considered analytic solutions of the functional equation $ x + g(y + f(x)) = y + g(x + f(y)) $ on the real line. Solutions in the class of twice differentiable functions are given in the author’s paper Differentiable solutions of a functional equation related to the non-power means (Aequationes Math. 55 (1998)). During the 38th International Symposium on Functional Equations N. Brillouet-Belluot presented the proof that differentiable solutions of this equation have the same form as in the previous cases. We present solutions of the equation in the class of monotonic Jensen convex or Jensen concave functions on the real line. This time we get already some new families of solutions.