Asymptotically logarithmic power series for $z\to 1$ with positive coefficients

Question

Consider a power series $S(z)=\sum_{i=1}^{\infty} a_i z^i$ which is analytic for $|z|<1$ and has non-negative coefficients $a_i$. If I know that for $q>p>0$ and $z\in (1-\varepsilon, 1)$ it holds $$ p(-\log(1-z))\leq S(z) \leq q(-\log(1-z)),$$ then it follows immediatley, that $$ S(z)\sim -\log(1-z)$$ as $z\to 1$. Motivated by Analytic Combinatorics, Flajolet, Sedgewick I am wondering what I can say about the behaviour of the sequence $(a_i)_{i\in\mathbb{N}}$. I guess it should behave like $a_i\sim i^{-1}$ as $i\to\infty$, since otherwise the lower or upper bound should not be satisfied, but I do not succeed at doing a formal proof. How can I make this conclusion (if it is correct)?