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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Migration of zeros for successive derivatives of entire functions

Author(s): Arie Harel; Su Namn; Jacob Sturm
Journal: Proc. Amer. Math. Soc. 127 (1999), 563-567.
MSC (1991): Primary 30A66
MathSciNet review: 1468192
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Abstract | References | Similar articles | Additional information

Abstract: It is shown that if $f$ is an entire function of order less than one, all of whose zeros are real, then the minimal root of $f^{(k)}$ is an increasing function of $k$ which accelerates as $k$ increases.

References:

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L. Ahlfors, Complex Analysis: An introduction to the theory of analytic functions of one complex variable, McGraw-Hill, (1966). MR 32:5844

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R.P. Boas, Jr., Zeros of the successive derivates of entire functions, J. of Math. Analysis and Applications 42 (1973), 446-473. MR 48:11508

[CCS]
T. Craven, G. Csordas and W. Smith, The zeros of derivatives of entire functions and the Pólya-Wiman Conjecture, Ann. of Math. 125 (1987), 405-431. MR 88a:30007

[G]
M.W. Gontcharoff, Recherches sur les derivees successives des fonctions analytiques, Ann. Sci. de L'Eclole Norm. Sup. Math 47 (1930) pp 1-92.

[HW1]
S. Hellerstein and J. Williamson, Derivatives of entire functions and a question of Pólya, Trans. of the Amer. Math. Soc. 227 (1977), 227-249. MR 55:8353

[HW2]
S. Hellerstein and J. Williamson, Derivatives of entire functions and a question of Pólya II, Trans. of the Amer. Math. Soc. 234 (1977), 497-503. MR 58:1151

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G. Pólya, On the zeros of the derivatives of a function and its analytic character, Bull. Amer. Math. Soc 49 (1943), 178-191. MR 4:192d

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M. S. Wilf, Budan's theorem for a class of entire functions, Proc. Amer. Math. Soc. 13 (1962), 122-125. MR 24:A3290

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Additional Information:

Arie Harel
Affiliation: Department of Mathematics, National University of Singapore, Lower Kent Ridge Road, Singapore 119260
Email: harel@math.nus.edu.sg

Su Namn
Affiliation: Department of Management Information Systems, Hannam University, Taejon, Korea
Email: namn@eve.hannam.ac.kr

Jacob Sturm
Affiliation: Department of Mathematics and Computer Science, Rutgers University, Newark, New Jersey 07102
Email: sturm@andromeda.rutgers.edu

DOI: 10.1090/S0002-9939-99-04542-6
PII: S 0002-9939(99)04542-6
Keywords: Successive derivatives, entire functions, migration
Received by editor(s): December 29, 1996
Received by editor(s) in revised form: March 24, 1997 and June 4, 1997
Communicated by: Albert Baernstein II
Copyright of article: Copyright 1999, American Mathematical Society




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