Zeros of the successive derivatives of Hadamard gap series

Author:
Robert M. Gethner

Journal:
Trans. Amer. Math. Soc. **339** (1993), 799-807

MSC:
Primary 30D35; Secondary 30B10

DOI:
https://doi.org/10.1090/S0002-9947-1993-1123453-3

MathSciNet review:
1123453

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Abstract | References | Similar Articles | Additional Information

Abstract: A complex number is in the *final set* of an analytic function , as defined by Pólya, if every neighborhood of contains zeros of infinitely many . If is a Hadamard gap series, then the part of the final set in the open disk of convergence is the origin along with a union of concentric circles.

**[1]**A. Edrei and G. R. Maclane,*On the zeroes of the derivatives of an entire function*, Proc. Amer. Math. Soc.**8**(1957), 702-706. MR**0087741 (19:403a)****[2]**R. M. Gethner,*Zeros of the successive derivatives of Hadamard gap series in the unit disk*, Michigan Math. J.**36**(1989), 403-414. MR**1027076 (91b:30004)****[3]**W. K. Hayman,*Angular value distribution of power series with gaps*, Proc. London Math. Soc. (3)**24**(1972), 590-624. MR**0306497 (46:5623)****[4]**G. Pólya,*Über die Nullstellen sukzessiver Derivierten*, Collected Papers (R. P. Boas, ed.), MIT Press, Cambridge, Mass., 1974.**[5]**-,*On the zeros of the derivatives of a function and its analytic character*, Collected Papers (R. P. Boas, ed.), MIT Press, Cambridge, Mass., 1974.**[6]**G. Pólya and G. Szegö,*Problems and theorems in analysis*, vol. 2, Springer-Verlag, New York, 1972.**[7]**E. C. Titchmarsh,*The theory of functions*, 2nd ed., Oxford Univ. Press, Oxford, 1939. MR**0197687 (33:5850)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1123453-3

Keywords:
Final set,
successive derivatives,
Hadamard gaps,
gap series

Article copyright:
© Copyright 1993
American Mathematical Society