Taylor Domination, Turán lemma, and Poincaré-Perron Sequences

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 Turán's inequality on the sums of powers. Next we consider linear recurrence relations of the Poincaré type $a_{k}=\sum_{j=1}^{d}[c_{j}+ψ_{j}(k)]a_{k-j},\ \ k=d,d+1,\dots,\quad\text{with }\lim_{k\rightarrow\infty}ψ_{j}(k)=0.$ 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.

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