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2013 | 11 | 9 | 1643-1650
Tytuł artykułu

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

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EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
11
Numer
9
Strony
1643-1650
Opis fizyczny
Daty
wydano
2013-09-01
online
2013-06-28
Twórcy
Bibliografia
  • [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
  • [3] Craven T., Csordas G., Iterated Laguerre and Turán inequalities, JIPAM. J. Inequal. Pure Appl. Math., 2002, 3, #39
  • [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
  • [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
  • [6] Goldberg A.A., Ostrovskii I.V., Value Distribution of Meromorphic Functions, Transl. Math. Monogr., 236, American Mathematical Society, Providence, 2008
  • [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
  • [8] Koosis P., The Logarithmic Integral I, Cambridge Stud. Adv. Math., 12, Cambridge University Press, Cambridge, 1988 http://dx.doi.org/10.1017/CBO9780511566196
  • [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
  • [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
  • [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
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-013-0269-x
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