Problems similar to the additive divisor problem

For multiplicative functions $f(n)$, let the following conditions be satisfied: $f(n)\ge0$, $f(p^r)\le A^r$, $A>0$, and for any $\varepsilon>0$ there exist constants $A_\varepsilon$, $\alpha>0$ such that $f(n)\le A_\varepsilon n^\varepsilon$ and $\sum_{p\le x}f(p)\ln p\ge\alpha x$. For such functions, the following relation is proved:
$$ \sum_{n\le x}f(n)\tau(n-1)=C(f)\sum_{n\le x}f(n)\ln x\bigl(1+o(1)\bigr). $$
Here $\tau(n)$ is the number of divisors of $n$ and $C(f)$ is a constant.