Functional equations involving means of functions on the complex plane (Q1271580)

The author offers the continuous solutions, mapping \(\mathbb{C}\) into itself, of the functional equation \[ (1/N)\sum_{n=0}^{N} f(z+\omega^{n}\zeta)=f(z)+g(z)h(\zeta) \] (\(N\geq 2;\omega\) is a primitive \(N\)-th root of unity) and its integral analog, which generalize several known and important functional equations. One of the remarkable statements is that for \(g=1, h=f\) the general such solution is given by \(f(z)=az^{n}+b\bar{z}^{n}+c| z| ^{2}\) (notice the difference between the exponents if \(N>2\)). Several particular cases are discussed.