Taylor Domination, Difference Equations, and Bautin Ideals

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.