Please use this identifier to cite or link to this item: http://repositorio.unicamp.br/jspui/handle/REPOSIP/68391
Type: Artigo de periódico
Title: HIGHER ORDER TURAN INEQUALITIES FOR THE RIEMANN xi-FUNCTION
Author: Dimitrov, DK
Lucas, FR
Abstract: The simplest necessary conditions for an entire function psi(x) = Sigma(infinity)(k=0) gamma k xk/k! to be in the Laguerre-Polya class are the Turan inequalities gamma(2)(k) - gamma k+1 gamma k-1 >= 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 Turan inequalities 4(gamma(2)(n) - gamma n-1 gamma n+1)(gamma(2)(n+1), - gamma n gamma n+2) - (gamma n gamma n+1 - gamma n-1 gamma n+2)(2) >= 0 are also necessary conditions for a function of the above form to belong to the Laguerre-Polya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic. Polya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turan inequalities hold for the coefficients of the Riemann xi-function. In this short paper, we prove that the higher order Turan inequalities also hold for the xi-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three.
Subject: Laguerre-Polya class
Maclaurin coefficients
Turan inequalities
Jensen polynomials
Riemann xi function
Country: EUA
Editor: Amer Mathematical Soc
Rights: aberto
Identifier DOI: 10.1090/S0002-9939-2010-10515-4
Date Issue: 2011
Appears in Collections:Unicamp - Artigos e Outros Documentos

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