The generalized Laguerre inequalities and functions in the Laguerre-Pólya class

George Csordas; Anna Vishnyakova

Open Mathematics (2013)

  • Volume: 11, Issue: 9, page 1643-1650
  • ISSN: 2391-5455

Abstract

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The principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 concludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent, ingenious extension of the Laguerre-type inequalities.

How to cite

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George Csordas, and Anna Vishnyakova. "The generalized Laguerre inequalities and functions in the Laguerre-Pólya class." Open Mathematics 11.9 (2013): 1643-1650. <http://eudml.org/doc/269318>.

@article{GeorgeCsordas2013,
abstract = {The principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 concludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent, ingenious extension of the Laguerre-type inequalities.},
author = {George Csordas, Anna Vishnyakova},
journal = {Open Mathematics},
keywords = {Laguerre-Pólya class; Generalized Laguerre-type inequalities; generalized Laguerre-type inequalities},
language = {eng},
number = {9},
pages = {1643-1650},
title = {The generalized Laguerre inequalities and functions in the Laguerre-Pólya class},
url = {http://eudml.org/doc/269318},
volume = {11},
year = {2013},
}

TY - JOUR
AU - George Csordas
AU - Anna Vishnyakova
TI - The generalized Laguerre inequalities and functions in the Laguerre-Pólya class
JO - Open Mathematics
PY - 2013
VL - 11
IS - 9
SP - 1643
EP - 1650
AB - The principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 concludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent, ingenious extension of the Laguerre-type inequalities.
LA - eng
KW - Laguerre-Pólya class; Generalized Laguerre-type inequalities; generalized Laguerre-type inequalities
UR - http://eudml.org/doc/269318
ER -

References

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  8. [8] Koosis P., The Logarithmic Integral I, Cambridge Stud. Adv. Math., 12, Cambridge University Press, Cambridge, 1988 http://dx.doi.org/10.1017/CBO9780511566196 Zbl0665.30038
  9. [9] Levin B.Ja., Distribution of Zeros of Entire Functions, Transl. Math. Monogr., 5, American Mathematical Society, Providence, 1980 
  10. [10] Obreschkoff N., Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963 Zbl0156.28202
  11. [11] Patrick M.L., Extensions of inequalities of the Laguerre and Turán type, Pacific J. Math., 1973, 44(2), 675–682 http://dx.doi.org/10.2140/pjm.1973.44.675 Zbl0265.33012
  12. [12] Pólya G., Collected Papers, II, Mathematicians of Our Time, 8, MIT Press, Cambridge, 1974 
  13. [13] Pólya G., Schur J., Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math., 1914, 144, 89–113 Zbl45.0176.01

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