On asymptotic behaviour of Dirichlet inverse

Let $f(n)$ be an arithmetic function with $f(1)\mathrm{\ne }0$ and let $f^{-1}(n)$ be its reciprocal with respect to the Dirichlet convolution. We study the asymptotic behaviour of $\mathrm{\mid }{f}^{-1}(n)\mathrm{\mid }$ with regard to the asymptotic behaviour of $\mathrm{\mid }f(n)\mathrm{\mid }$ assuming that the latter one grows or decays with at most polynomial or exponential speed. As a by-product, we obtain simple but constructive upper bounds for the number of ordered factorizations of $n$ into $k$ factors.