I'm interested in Singularity Analysis after reading this Math.Stack post. In the "Analytic Combinatorics" book by P. Flajolet and R. Sedgewick, at p.390, the following transfer theorem (Theorem VI.3) is stated :
Let $f(z)$ be a $\Delta$-analytic function at $1$ obeying the following asymptotic behaviour near $z = 1$
$$f(z) = \mathcal{O} \left( (1-z)^{-\alpha} \log \left(\frac{1}{1-z} \right)^{\beta} \right)$$
Then the coefficients $f_n$ of the Taylor expansion of $f(z)$ at $z = 0$ satisfy
$$f_n = \mathcal{O} \left( n^{\alpha-1} \log(n)^{\beta} \right)$$
and there are similar results for little-o or sim-relations.
My question is the following : is the converse statement true ? Assuming that $f(z)$ is $\Delta$-analytic at $1$, does the asymptotic growth of the $f_n$ tell us what is the worst singularity we can expect near $z = 1$ ?
References concerning Singularity Analysis are welcomed !