Functional equations involving means of functions on the complex plane

Abstract

We find the continuous solutions f, g, h of the functional equation¶¶ $ {1\over N} \sum^{N - 1}_{n = 0} f(z + \omega^n\zeta) = f(z) + g(z) h(\zeta), \qquad z,\zeta \in \Bbb {C}, $¶for any primitive N ^th root $\omega$ of unity, and of a similar one in which the sum is replaced by an integral over the unit circle. Particular cases are the quadratic functional equation and the functional equation of symmetric second differences in product form in which N = 2.