Zeros of partial sums and remainders of power series

Abstract

For a power series f ( z ) = Σ k = 0 ∞ a k z k f(z) = \Sigma _{k = 0}^\infty {a_k}{z^k} let s n ( f ) {s_n}(f) denote the maximum modulus of the zeros of the nth partial sum of f and let r n ( f ) {r_n}(f) denote the smallest modulus of a zero of the nth normalized remainder Σ k = n ∞ a k z k − n \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 ) } \{ {s_n}(f)\} and { r n ( f ) } \{ {r_n}(f)\} . The principal growth measure used is that of R-type: if R = { R n } R = \{ {R_n}\} is a nondecreasing sequence of positive numbers such that lim ( R n + 1 / R n ) = 1 \lim ({R_{n + 1}}/{R_n}) = 1 , then the R-type of f is τ R ( f ) = lim sup | a n R 1 R 2 ⋯ R n | 1 / n {\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 \[ τ R ( f ) lim inf ( s n ( f ) / R n ) ≦ P and τ R ( f ) lim sup ( r n ( f ) / R n ) ≧ ( 1 / P ) {\tau _R}(f)\lim \inf ({s_n}(f)/{R_n}) \leqq P\quad {\text {and}}\quad {\tau _R}(f)\lim \sup ({r_n}(f)/{R_n}) \geqq (1/P) \] 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.