The Laguerre inequality and the distribution of zeros of entire functions

George Csordas[1]; Alain Escassut[2]

  • [1] University of Hawaii Department of Mathematics Honolulu, HI 96822 USA
  • [2] Université Blaise Pascal Laboratoire de Mathematiques Les Cézaux 63177 Aubiere Cedex FRANCE

Annales mathématiques Blaise Pascal (2005)

  • Volume: 12, Issue: 2, page 331-345
  • ISSN: 1259-1734

Abstract

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The Laguerre inequality and the distribution of zeros of real entire functions are investigated with the aid of certain infinite-order differential operators. The paper includes new proofs, problems, conjectures and many illustrative examples and counterexamples.

How to cite

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Csordas, George, and Escassut, Alain. "The Laguerre inequality and the distribution of zeros of entire functions." Annales mathématiques Blaise Pascal 12.2 (2005): 331-345. <http://eudml.org/doc/10523>.

@article{Csordas2005,
abstract = {The Laguerre inequality and the distribution of zeros of real entire functions are investigated with the aid of certain infinite-order differential operators. The paper includes new proofs, problems, conjectures and many illustrative examples and counterexamples.},
affiliation = {University of Hawaii Department of Mathematics Honolulu, HI 96822 USA; Université Blaise Pascal Laboratoire de Mathematiques Les Cézaux 63177 Aubiere Cedex FRANCE},
author = {Csordas, George, Escassut, Alain},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Laguerre inequality; Laguerre-Pólya class; infinite order differential operator; Hermite-Biehler theorem},
language = {eng},
month = {7},
number = {2},
pages = {331-345},
publisher = {Annales mathématiques Blaise Pascal},
title = {The Laguerre inequality and the distribution of zeros of entire functions},
url = {http://eudml.org/doc/10523},
volume = {12},
year = {2005},
}

TY - JOUR
AU - Csordas, George
AU - Escassut, Alain
TI - The Laguerre inequality and the distribution of zeros of entire functions
JO - Annales mathématiques Blaise Pascal
DA - 2005/7//
PB - Annales mathématiques Blaise Pascal
VL - 12
IS - 2
SP - 331
EP - 345
AB - The Laguerre inequality and the distribution of zeros of real entire functions are investigated with the aid of certain infinite-order differential operators. The paper includes new proofs, problems, conjectures and many illustrative examples and counterexamples.
LA - eng
KW - Laguerre inequality; Laguerre-Pólya class; infinite order differential operator; Hermite-Biehler theorem
UR - http://eudml.org/doc/10523
ER -

References

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  1. R.P. Boas, Entire Functions, (1954), Academic Press, New York Zbl0058.30201MR68627
  2. D. A. Cardon, S. A. de Gaston, Differential operators and entire functions with simple real zeros, J. Math. Anal. Appl. 301 (2005), 286-393 Zbl1066.30031MR2105679
  3. T. Craven, G. Csordas, Differential operators of infinite-order and the distribution of zeros of entire functions, J. Math. Anal. Appl. 186 (1994), 799-820 Zbl0810.30006MR1293856
  4. G. Csordas, R. S. Varga, Necessary and sufficient conditions and the Riemann Hypothesis, Adv. in Appl. Math. 11 (1990), 328-357 Zbl0707.11062MR1061423
  5. N.G. de Bruijn, The roots of Trigonometric integrals, Duke Math 7 (1950), 197-226 Zbl0038.23302MR37351
  6. W. Smith G. Csordas, R. S. Varga, Level sets of real entire functions and the Laguerre inequalities, Analysis 12 (1992), 377-402 Zbl0759.30016MR1182637
  7. J. L. W. V. Jensen, Recherches sur la théorie des équations, Acta Math. 36 (1913), 181-195 Zbl43.0158.01MR1555086
  8. L. Kuipers, Note on the location of zeros of polynomials, Simon Stevin 31 (1957), 61-72 Zbl0082.02203MR83970
  9. B. Ja. Levin, Distribution of Zeros of Entire Functions, (1980), Transl. Math. Mono. Vol 5. Amer. Math. Soc, Providence, R.I. MR589888
  10. N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, (1963), VEB Deutscher Verlag der Wissenschaften, Berlin Zbl0156.28202MR164003
  11. G. Pólya, Über die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. V. Jensen, Klg. Danske Vid. Sel. Math.-Fys. Medd 7 (1927), 3-33 
  12. G. Pólya, Location of Zeros, (1974), Collected papers, Vol.2, R.P. Boas, MIT Press, Cambridge, MA MR505093
  13. P. C. Sikkema, Differential Operators and Differential Equations of Infinite Order with Constant Coefficients, (1953), P. Noordhoff N.V., Groningen Zbl0051.34402MR60082

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