  We compare three approaches to studying the behavior of an analytic function
$f(z)=\sum_{k=0}^\infty a_kz^k$ from its Taylor coefficients. The first is
"Taylor domination" property for $f(z)$ in the complex disk $D_R$, which is an
inequality of the form \[ |a_{k}|R^{k}\leq C\ \max_{i=0,\dots,N}\ |a_{i}|R^{i},
\ k \geq N+1. \] The second approach is based on a possibility to generate
$a_k$ via recurrence relations. Specifically, we consider linear non-stationary
recurrences of the form \[ a_{k}=\sum_{j=1}^{d}c_{j}(k)\cdot a_{k-j},\ \
k=d,d+1,\dots, \] with uniformly bounded coefficients.
  In the third approach we assume that $a_k=a_k(\lambda)$ are polynomials in a
finite-dimensional parameter $\lambda \in {\mathbb C}^n.$ We study "Bautin
ideals" $I_k$ generated by $a_{1}(\lambda),\ldots,a_{k}(\lambda)$ in the ring
${\mathbb C}[\lambda]$ of polynomials in $\lambda$.
  \smallskip
  These three approaches turn out to be closely related. We present some
results and questions in this direction.
