Taylor Domination, Tur\'an lemma, and Poincar\'e-Perron Sequences

Abstract

We consider "Taylor domination" property for an analytic function $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k},$ in the complex disk $D_R$, which is an inequality of the form \[ |a_{k}|R^{k}\leq C\ \max_{i=0,\dots,N}\ |a_{i}|R^{i}, \ k \geq N+1. \] This property is closely related to the classical notion of "valency" of $f$ in $D_R$. For $f$ - rational function we show that Taylor domination is essentially equivalent to a well-known and widely used Tur\'an's inequality on the sums of powers. Next we consider linear recurrence relations of the Poincar\'e type \[ a_{k}=\sum_{j=1}^{d}[c_{j}+\psi_{j}(k)]a_{k-j},\ \ k=d,d+1,\dots,\quad\text{with }\lim_{k\rightarrow\infty}\psi_{j}(k)=0. \] We show that the generating functions of their solutions possess Taylor domination with explicitly specified parameters. As the main example we consider moment generating functions, i.e. the Stieltjes transforms \[ S_{g}\left(z\right)=\int\frac{g\left(x\right)dx}{1-zx}. \] We show Taylor domination property for such $S_{g}$ when $g$ is a piecewise D-finite function, satisfying on each continuity segment a linear ODE with polynomial coefficients.