On the Connection between Gaps in Power Series and the Roots of Their Partial Sums

be a power series with the radius of convergence 1. We say that it has Ostrowski gaps p if there exists a pi, such that | an \ < pn for mk á « Û «*. It has infinite Ostrowski gaps p (pp there corresponds a pair of infinite sequences mk and nk (depending on p') with mk<nk and lim nk/mk= « such that | a» | < p'B for mk á « ú nk. We denote by.d(«, r) the number of roots of f(x) = 1 -faix + • • • + anxn within the circle of radius r. It is well known that every overconvergent power series has Ostrowski gaps, and that every power series with Ostrowski gaps is overconvergent in a domain of which every regular point of the circle of convergence is an interior point. We are going to prove the following theorems: