Higher order turán inequalities for the Riemann ξ-function

dc.contributor.authorDimitrov, Dimitar Kolev [UNESP]
dc.contributor.authorLucas, Fábio R.
dc.contributor.institutionUniversidade Estadual Paulista (Unesp)
dc.contributor.institutionUniversidade Estadual de Campinas (UNICAMP)
dc.date.accessioned2014-05-27T11:25:28Z
dc.date.available2014-05-27T11:25:28Z
dc.date.issued2011-03-01
dc.description.abstractThe simplest necessary conditions for an entire function ψ(x) =∞ ∑ k=0 γk xk/k! to be in the Laguerre-Pólya class are the Turán inequalities γ2 k- γk+1γk-1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(γ2 n - γn-1γn+1)(γ2n +1 - γnγn+2) - (γnγn+1 - γn-1γn+2) 2 ≥ 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 ψ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ψ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. © 2010 American Mathematical Society.en
dc.description.affiliationDepartamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP
dc.description.affiliationDepartamento de matemática Aplicada IMECC UNICAMP, 13083-859 Campinas, SP
dc.description.affiliationUnespDepartamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP
dc.description.sponsorshipFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
dc.description.sponsorshipConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
dc.description.sponsorshipCoordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
dc.description.sponsorshipIdFAPESP: 03/01874-2
dc.description.sponsorshipIdFAPESP: 06/60420-0
dc.description.sponsorshipIdCNPq: 305622/2009-9
dc.description.sponsorshipIdCAPES: DGU-160
dc.format.extent1013-1022
dc.identifierhttp://dx.doi.org/10.1090/S0002-9939-2010-10515-4
dc.identifier.citationProceedings of the American Mathematical Society, v. 139, n. 3, p. 1013-1022, 2011.
dc.identifier.doi10.1090/S0002-9939-2010-10515-4
dc.identifier.file2-s2.0-79951846250.pdf
dc.identifier.issn0002-9939
dc.identifier.lattes1681267716971253
dc.identifier.scopus2-s2.0-79951846250
dc.identifier.urihttp://hdl.handle.net/11449/132295
dc.identifier.wosWOS:000288727900024
dc.language.isoeng
dc.publisherAmer Mathematical Soc
dc.relation.ispartofProceedings of the American Mathematical Society
dc.relation.ispartofjcr0.707
dc.relation.ispartofsjr1,183
dc.rights.accessRightsAcesso aberto
dc.sourceScopus
dc.subjectJensen polynomials
dc.subjectLaguerre-Pólya class
dc.subjectMaclaurin coefficients
dc.subjectRiemann ξ function
dc.subjectTurán inequalities
dc.titleHigher order turán inequalities for the Riemann ξ-functionen
dc.typeArtigo
dcterms.licensehttp://www.ams.org/publications/authors/ctp
dcterms.rightsHolderAmer Mathematical Soc
unesp.author.lattes1681267716971253
unesp.campusUniversidade Estadual Paulista (Unesp), Instituto de Biociências Letras e Ciências Exatas, São José do Rio Pretopt
unesp.departmentCiências da Computação e Estatística - IBILCEpt

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