On multiplicative functions which are small on average

Let $f$ be a completely multiplicative function that assumes values inside the unit disc. We show that if ${\sum }_{n<x}f(n)\ll x\mathrm{/}(\log x{)}^A$, $x>2$, for some $A>2$, then either $f(p)$ is small on average or $f$ pretends to be $\mu (n)n^{it}$ for some $t$.