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- W2080846623 abstract "For a power series f(z) = o a_kZk let sn(f) denote the maximum modulus of the zeros of the nth partial sum of f and let rn(f) denote the smallest modulus of a zero of the nth normalized remainder >k=n akZk' -n. The present paper investigates the relationships between the growth of the analytic function f and the behavior of the sequences {sn(f)} and {rn(f)}. The principal growth measure used is that of R-type: if R = {Rn} is a nondecreasing sequence of positive numbers such that lim (Rn+ 1Rn) = 1, then the R-type off is TR(f)=lim sup IanRlR2 . RnlIn. We prove that there is a constant P such that -TR(f) lim inf (sn(f)lRn) _ P and TR(f) lim sup (rn(f)/Rn) > (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." @default.
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- W2080846623 date "1972-01-01" @default.
- W2080846623 modified "2023-10-18" @default.
- W2080846623 title "Zeros of partial sums and remainders of power series" @default.
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- W2080846623 doi "https://doi.org/10.1090/s0002-9947-1972-0299762-3" @default.
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