# 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

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topGeorge 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

top- [1] Boas R.P. Jr., Entire Functions, Academic Press, New York, 1954
- [2] Cardon D.A., Extended Laguerre inequalities and a criterion for real zeros, In: Progress in Analysis and its Applications, London, July 13–18, 2009, World Scientific, Hackensack, 2010, 143–149 Zbl1262.30004
- [3] Craven T., Csordas G., Iterated Laguerre and Turán inequalities, JIPAM. J. Inequal. Pure Appl. Math., 2002, 3, #39 Zbl1007.30007
- [4] Csordas G., Varga R.S., Necessary and sufficient conditions and the Riemann Hypothesis, Adv. in Appl. Math., 1990, 11(3), 328–357 http://dx.doi.org/10.1016/0196-8858(90)90013-O Zbl0707.11062
- [5] Dilcher K., Stolarsky K.B., On a class of nonlinear differential operators acting on polynomials, J. Math. Anal. Appl., 1992, 170(2), 382–400 http://dx.doi.org/10.1016/0022-247X(92)90025-9 Zbl0768.30005
- [6] Goldberg A.A., Ostrovskii I.V., Value Distribution of Meromorphic Functions, Transl. Math. Monogr., 236, American Mathematical Society, Providence, 2008 Zbl1152.30026
- [7] Jensen J.L.W.V., Recherches sur la théorie des équations, Acta Math., 1913, 36(1), 181–195 http://dx.doi.org/10.1007/BF02422380 Zbl43.0158.01
- [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] Levin B.Ja., Distribution of Zeros of Entire Functions, Transl. Math. Monogr., 5, American Mathematical Society, Providence, 1980
- [10] Obreschkoff N., Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963 Zbl0156.28202
- [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] Pólya G., Collected Papers, II, Mathematicians of Our Time, 8, MIT Press, Cambridge, 1974
- [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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