ON A FUNCTIONAL EQUATION APPEARING ON THE MARGINS OF A MEAN INVARIANCE PROBLEM

Given a continuous strictly monotonic real-valued function alpha, defined on an interval I, and a function omega: I -> (0, +infinity) we denote by B-omega(alpha) the Bajraktarevic mean generated by alpha and weighted by omega: B-omega(alpha)(x, y) = alpha(-1) (omega(x)/omega(x) + omega(y)alpha(x) + omega(y)/omega(x) + omega(y)alpha(y)), x, y epsilon I: We find a necessary integral formula for all possible three times differentiable solutions (phi, psi) of the functional equation r(x)B-s(phi) (x, y) + r(y)B-t(psi) (x, y) = r(x)x + r(y)y, where r, s, t : I -> (0, +infinity) are three times differentiable functions and the first derivatives of phi, psi and r do not vanish. However, we show that not every pair (phi, psi) given by the found formula actually satisfies the above equation.