On the pairs of multiplicative functions satisfying some relations

It is proved that if f and g are complex-valued multiplicative functions such that $ g(2n + 1) - Af(n) \to 0 (n \to \infty) $ , for some $ A \neq 0 $ , then either $ f(n) \to 0 (n \to \infty) $ , or $ f(n) = n^s $ , $ 0 \le {\rm Re} s < 1 $ and $ A = f(2), g(n) = f(n) $ for every odd n.