On coefficients and zeros of sections of power series

Shaw, J. K.

doi:10.1090/S0002-9939-1973-0318461-9

On coefficients and zeros of sections of power series
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by J. K. Shaw

Proc. Amer. Math. Soc. 39 (1973), 567-570

DOI: https://doi.org/10.1090/S0002-9939-1973-0318461-9

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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 $|{a_n}{|^{1/n}}{s_n}$ 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.

References

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J. D. Buckholtz and J. K. Shaw, Zeros of partial sums and remainders of power series, Trans. Amer. Math. Soc. 166 (1972), 269–284. MR 299762, DOI 10.1090/S0002-9947-1972-0299762-3
Jacob Korevaar, The zeros of approximating polynomials and the canonical representation of an entire function, Duke Math. J. 18 (1951), 573–592. MR 43203
S. M. Shah, The maximum term of an entire series, Math. Student 10 (1942), 80–82. MR 7434