Taylor domination, Turán lemma, and Poincaré-Perron sequences

We consider “Taylor domination” property for an analytic function (Formula presented), in the complex disk D_R, which is an inequality of the form (Formula presented). 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 (Formula presented) 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 (Formula presented). 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.