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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Zeros of partial sums and remainders of power series


Authors: J. D. Buckholtz and J. K. Shaw
Journal: Trans. Amer. Math. Soc. 166 (1972), 269-284
MSC: Primary 30A08
MathSciNet review: 0299762
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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 \vert{a_n}{R_1}{R_2} \cdots {R_n}{\vert^{1/n}}$. We prove that there is a constant P such that

$\displaystyle {\tau _R}(f)\lim \inf ({s_n}(f)/{R_n}) \leqq P\quad {\text{and}}\quad {\tau _R}(f)\lim \sup ({r_n}(f)/{R_n}) \geqq (1/P)$

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.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0299762-3
PII: S 0002-9947(1972)0299762-3
Keywords: The power series constant, zeros of partial sums, zeros of remainders, R-type, entire functions, extremal functions
Article copyright: © Copyright 1972 American Mathematical Society