Taylor series with limit-points on a finite number of circles

Abstract:

Let $S(z):\sum \nolimits _{n = 0}^\infty {{a_n}{z_n}}$ be a power series with complex coefficients. For each $z$ in the unit circle $T = \{ z \in \mathbb {C}:|z| = 1\}$ we denote by $L(z)$ the set of limit-points of the sequence $\{ {s_n}(z)\}$ of the partial sums of $S(z)$. In this paper we examine Taylor series for which the set $L(z)$, for $z$ in an infinite subset of $T$, is the union of a finite number, uniformly bounded in $z$, of concentric circles. We show that, if in addition $\lim \inf |{a_n}|\; > 0$, a complete characterization of these series in terms of their coefficients is possible (see Theorem 1).