Existence of nontrivial solutions of linear functional equations.

In this paper we study the functional equation where a, b, c are fixed complex numbers and $${f \colon \mathbb{C} \to \mathbb{C}}$$ is the unknown function. We show, that if there is i such that $${b_i / c_i \neq b_j /c_j}$$ holds for any $${1 \leq j \leq n,\ j \neq i}$$ , the functional equation has a nonconstant solution if and only if there are field automorphisms $${\phi_1, \ldots, \phi_k}$$ of $${\mathbb{C}}$$ such that $${\phi_1 \cdots \phi_k}$$ is a solution of the equation. [ABSTRACT FROM AUTHOR]