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 let denote the maximum modulus of the zeros of the *n*th partial sum of *f* and let denote the smallest modulus of a zero of the *n*th 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 *R*-type: if is a nondecreasing sequence of positive numbers such that , then the *R*-type of *f* is . We prove that there is a constant *P* such that

*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.

**[1]**Ralph P. Boas Jr. and R. Creighton Buck,*Polynomial expansions of analytic functions*, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. Heft 19, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR**0094466****[2]**J. D. Buckholtz,*Zeros of partial sums of power series*, Michigan Math. J.**15**(1968), 481–484. MR**0235097****[3]**J. D. Buckholtz,*Zeros of partial sums of power series. II*, Michigan Math. J.**17**(1970), 5–14. MR**0259076****[4]**J. D. Buckholtz and J. L. Frank,*Whittaker constants*, Proc. London Math. Soc. (3)**23**(1971), 348–370. MR**0296297****[5]**M. B. Porter,*On the polynomial convergents of a power series*, Ann. of Math. (2)**8**(1907), no. 4, 189–192. MR**1502347**, 10.2307/1967824**[6]**M. Tsuji,*On the distribution of the zero points of sections of a power series*. III, Japan. J. Math.**3**(1926), 49-51.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0299762-3

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