On the general solution of the triangle mean value equation (Q788218)

The general solution of the functional equation \[ f(x-t,y- \frac{t}{\sqrt{3}})+f(x+t,y- \frac{t}{\sqrt{3}})+f(x,y+\frac{2t}{\sqrt{3}})=3f(x,y) \] is obtained here. The form of the solution is rather involved. The situation is considerably simpler in the case where f is assumed bounded on a set of positive Lebesgue measure; in this case it was shown by \textit{D. Z. Djoković} [Triangle functional equations and its generalization, Univ. Beograd. Publ. Elektroteh. Fak. Ser. Mat. Fiz. 181-196, 47-52 (1967)] that f is a linear combination of the harmonic polynomials \(x^ 3-3xy^ 2\), \(x^ 2-y^ 2\), xy, xy, x, y and 1.