Zeros of partial sums and remainders of power series
Authors:
J. D. Buckholtz and J. K. Shaw
Journal:
Trans. Amer. Math. Soc. 166 (1972), 269284
MSC:
Primary 30A08
MathSciNet review:
0299762
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Abstract: For a power series let denote the maximum modulus of the zeros of the nth partial sum of f and let denote the smallest modulus of a zero of the nth normalized remainder . The present paper investigates the relationships between the growth of the analytic function f and the behavior of the sequences and . The principal growth measure used is that of Rtype: if is a nondecreasing sequence of positive numbers such that , then the Rtype of f is . We prove that there is a constant P such that for functions f of positive finite Rtype. The constant P cannot be replaced by a smaller number in either inequality; P is called the power series constant.
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 [1]
 R. P. Boas, Jr. and R. C. Buck, Polynomial expansions of analytic functions, Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 19, SpringerVerlag, Berlin, 1958. MR 20 #984. MR 0094466 (20:984)
 [2]
 J. D. Buckholtz, Zeros of partial sums of power series, Michigan Math. J. 15 (1968), 481484. MR 38 #3409. MR 0235097 (38:3409)
 [3]
 , Zeros of partial sums of power series. II, Michigan Math. J. 17 (1970), 514. MR 41 #3718. MR 0259076 (41:3718)
 [4]
 J. D. Buckholtz and J. L. Frank, Whittaker constants, Proc. London Math. Soc. 3 (1971), 348370. MR 0296297 (45:5358)
 [5]
 M. B. Porter, On the polynomial convergents of a power series, Ann. of Math. (2) 8 (19061907), 189192. MR 1502347
 [6]
 M. Tsuji, On the distribution of the zero points of sections of a power series. III, Japan. J. Math. 3 (1926), 4951.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197202997623
PII:
S 00029947(1972)02997623
Keywords:
The power series constant,
zeros of partial sums,
zeros of remainders,
Rtype,
entire functions,
extremal functions
Article copyright:
© Copyright 1972 American Mathematical Society
