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

TL;DR

This paper explores the Taylor domination property for analytic functions, relating it to valency and Turán's inequality, and applies it to solutions of Poincaré-type recurrence relations and moment generating functions of piecewise D-finite functions.

Contribution

It establishes a connection between Taylor domination, Turán's inequality, and valency, and demonstrates how these properties apply to solutions of Poincaré recurrences and certain moment generating functions.

Findings

Taylor domination is equivalent to Turán's inequality for rational functions.

Solutions of Poincaré-type recurrences exhibit Taylor domination with explicit parameters.

Moment generating functions of piecewise D-finite functions satisfy Taylor domination.

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.