The generalized Laguerre inequalities and functions in the Laguerre-Pólya class (original) (raw)

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 D. A. Cardon's recent, ingenious extension of the Laguerre-type inequalities.

FAQs

sparkles

AI

What are the generalized real Laguerre inequalities?add

The generalized real Laguerre inequalities are conditions that ensure a real entire function belongs to the Laguerre-Pólya class, as shown by Theorem 1.2, stating that if L_n(x) ≥ 0 for all n and x ∈ R, then ϕ(x) ∈ L -P.

How does the density of zeros relate to the Laguerre-Pólya class?add

The research highlights that the genus of ϕ1(x) in the representation ϕ(x) = e^{-ax^2}ϕ1(x) is critical, where ϕ1 has order 0 or 1, influencing the function's membership in the L -P class.

What implications do the Laguerre inequalities have on the zeros of functions?add

The findings indicate that if a real entire function satisfies the generalized real Laguerre inequalities, it must have only real zeros, as demonstrated in Theorem 2.3.

How do the complex Laguerre inequalities characterize functions in L -P?add

Theorem 3.1 establishes that a real entire function satisfies the complex Laguerre inequalities if and only if it belongs to the Laguerre-Pólya class, confirming the relationships between these inequalities.

What is the consequence of an entire function having even order but non-real zeros?add

The text suggests that despite being of even order, a real entire function with non-real zeros does not necessarily belong to the Laguerre-Pólya class, emphasizing the nuanced nature of function classifications.

Loading...

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

References (13)

  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, Progress in Analysis and its Applications, Proceedings of the 7th International Isaac Conference, 2009, 143-149
  3. Craven T., Csordas G., Iterated Laguerre and Turán inequalities, J. Inequal. Pure Appl. Math., 2002, 3, 39, 14 pp (electronic)
  4. Csordas G., Varga R. S., Necessary and sufficient conditions and the Riemann Hypothesis, Adv. in Appl. Math., 1990, 11, 328-357
  5. Dilcher K., Stolarsky K. B., On a class of nonlinear operators acting on polynomials, J. Math. Anal. Appl., 1992, 170, 382-400
  6. Goldberg A. A., Ostrovskii I. V., Value distribution of meromorphic functions, Transl. Math. Mono. Vol. 236, Amer. Math. Soc., Providence, RI, 2008
  7. Jensen J. L. W. V., Recherches sur la théorie des équations, Acta Math., 1913, 36, 181-195
  8. Koosis P., The Logarithmic Integral I, Cambridge Studies in Advanced Mathematics 12, Cambridge University Press, Cambridge, 1988
  9. Levin B. Ja., Distribution of Zeros of Entire Functions, Transl. Math. Mono. Vol. 5, Amer. Math. Soc., Providence, RI, 1964; revised ed. 1980
  10. Obreschkoff N., Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963
  11. Patrick M., Extensions of inequalities of the Laguerre and Turán type, Pacific J. Math., 1973, 44, 675-682
  12. Pólya G., Collected Papers, Vol. II, Location of Zeros, R. P. Boas, ed., MIT Press, Cambridge, MA, 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