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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On coefficients and zeros of sections of power series

Author: J. K. Shaw
Journal: Proc. Amer. Math. Soc. 39 (1973), 567-570
MSC: Primary 30A08
MathSciNet review: 0318461
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Abstract: For a power series $ f(z) = \sum {a_k}{z^k}$ let $ {s_n}$ denote the maximum modulus of the zeros of the $ n$th partial sum $ \sum\nolimits_0^n {{a_k}} {z^k}$. Asymptotic bounds on the sequence $ \vert{a_n}{\vert^{1/n}}{s_n}$ are obtained for both entire functions and functions with finite radii of convergence. These extend the previous results of J. D. Buckholtz and J. K. Shaw. Finally, conjectures regarding best possible asymptotic bounds are stated.

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Keywords: Coefficients, zeros of partial sums, the power series constant, entire functions
Article copyright: © Copyright 1973 American Mathematical Society

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