Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a power series $ f(z) = \Sigma _{k = 0}^\infty {a_k}{z^k}$ let $ {s_n}(f)$ denote the maximum modulus of the zeros of the nth partial sum of f and let $ {r_n}(f)$ denote the smallest modulus of a zero of the nth normalized remainder $ \Sigma _{k = n}^\infty {a_k}{z^{k - n}}$. The present paper investigates the relationships between the growth of the analytic function f and the behavior of the sequences $ \{ {s_n}(f)\} $ and $ \{ {r_n}(f)\} $. The principal growth measure used is that of R-type: if $ R = \{ {R_n}\} $ is a nondecreasing sequence of positive numbers such that $ \lim ({R_{n + 1}}/{R_n}) = 1$, then the R-type of f is $ {\tau _R}(f) = \lim \sup \vert{a_n}{R_1}{R_2} \cdots {R_n}{\vert^{1/n}}$. We prove that there is a constant P such that

$\displaystyle {\tau _R}(f)\lim \inf ({s_n}(f)/{R_n}) \leqq P\quad {\text{and}}\quad {\tau _R}(f)\lim \sup ({r_n}(f)/{R_n}) \geqq (1/P)$

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.

References [Enhancements On Off] (What's this?)

  • [1] R. P. Boas, Jr. and R. C. Buck, Polynomial expansions of analytic functions, Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 19, Springer-Verlag, 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), 481-484. MR 38 #3409. MR 0235097 (38:3409)
  • [3] -, Zeros of partial sums of power series. II, Michigan Math. J. 17 (1970), 5-14. MR 41 #3718. MR 0259076 (41:3718)
  • [4] J. D. Buckholtz and J. L. Frank, Whittaker constants, Proc. London Math. Soc. 3 (1971), 348-370. MR 0296297 (45:5358)
  • [5] M. B. Porter, On the polynomial convergents of a power series, Ann. of Math. (2) 8 (1906-1907), 189-192. MR 1502347
  • [6] M. Tsuji, On the distribution of the zero points of sections of a power series. III, Japan. J. Math. 3 (1926), 49-51.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30A08

Retrieve articles in all journals with MSC: 30A08

Additional Information

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

American Mathematical Society