Zeros of partial sums and remainders of power series

Abstract:

For a power series $f(z) = \Sigma _{k = 0}^\infty {a_k}{z^k}$ let ${s_n}(f)$ denote the maximum modulus of the zeros of the nth partial sum of f and let ${r_n}(f)$ denote the smallest modulus of a zero of the nth normalized remainder $\Sigma _{k = n}^\infty {a_k}{z^{k - n}}$. The present paper investigates the relationships between the growth of the analytic function f and the behavior of the sequences $\{ {s_n}(f)\}$ and $\{ {r_n}(f)\}$. The principal growth measure used is that of R-type: if $R = \{ {R_n}\}$ is a nondecreasing sequence of positive numbers such that $\lim ({R_{n + 1}}/{R_n}) = 1$, then the R-type of f is ${\tau _R}(f) = \lim \sup |{a_n}{R_1}{R_2} \cdots {R_n}{|^{1/n}}$. We prove that there is a constant P such that \[ {\tau _R}(f)\lim \inf ({s_n}(f)/{R_n}) \leqq P\quad {\text {and}}\quad {\tau _R}(f)\lim \sup ({r_n}(f)/{R_n}) \geqq (1/P)\] for functions f of positive finite R-type. The constant P cannot be replaced by a smaller number in either inequality; P is called the power series constant.