On the pairs of multiplicative functions with a
special relation

Abstract. It is proved that if \(f\) and \(g\) are complex-valued multiplicative functions satisfying \(g(An+1)-Cf(n)=o(1)\) as \(n\to \infty\) with some positive integer \(A\) and non-zero complex constant \(C\), then either \(f(n)=o(1),~g(An+1)=o(1)\) as \(n\to\infty\) or there exist a complex number \(s\) and functions \(F,~G\in {\cal M}\) such that \(f(n)=n^sF(n),~g(n)=n^sG(n)\), \((0\le {\rm Re}~ s < 1)\) and \(G(An+1)={1\over F(2)}F(n)\) are satisfied for all \(n\in {\Bbb N}\). All solutions of \((An+1)={1\over F(2)}F(n)\) are given.