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Higher order Turán inequalities for the Riemann $ \xi$-function

Authors: Dimitar K. Dimitrov and Fábio R. Lucas
Journal: Proc. Amer. Math. Soc. 139 (2011), 1013-1022
MSC (2010): Primary 33E20, 11M06; Secondary 26D07
Published electronically: September 15, 2010
MathSciNet review: 2745652
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Abstract: The simplest necessary conditions for an entire function

$ \displaystyle \psi(x) = \sum_{k=0}^{\infty} \gamma_{k} \frac{x^{k}}{k!} $

to be in the Laguerre-Pólya class are the Turán inequalities $ \gamma_k^2 - \gamma_{k+1} \gamma_{k-1} \geq 0$. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with $ \psi$ to be hyperbolic. The higher order Turán inequalities $ 4(\gamma_n^2-\gamma_{n-1}\gamma_{n+1})(\gamma_{n+1}^2 - \gamma_n\gamma_{n+2})-(\gamma_n\gamma_{n+1}-\gamma_{n-1}\gamma_{n+2})^2\geq 0$ are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic.

Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann $ \xi$-function. In this short paper, we prove that the higher order Turán inequalities also hold for the $ \xi$-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three.

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Additional Information

Dimitar K. Dimitrov
Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil

Fábio R. Lucas
Affiliation: Departamento de matemática Aplicada, IMECC, UNICAMP, 13083-859 Campinas, SP, Brazil

Keywords: Laguerre-Pólya class, Maclaurin coefficients, Turán inequalities, Jensen polynomials, Riemann $\xi$ function.
Received by editor(s): January 8, 2010
Received by editor(s) in revised form: March 5, 2010, and March 28, 2010
Published electronically: September 15, 2010
Additional Notes: Research supported by the Brazilian Science Foundations FAPESP under Grants 03/01874-2 and 06/60420-0, CNPq under Grant 305622/2009-9, and CAPES under Grant CAPES/DGU-160.
Communicated by: Walter Van Assche
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.