Higher order turán inequalities for the Riemann ξ-function
Autor(a) principal: | |
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Data de Publicação: | 2011 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UNESP |
Texto Completo: | http://dx.doi.org/10.1090/S0002-9939-2010-10515-4 http://hdl.handle.net/11449/132295 |
Resumo: | The 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. |
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Higher order turán inequalities for the Riemann ξ-functionJensen polynomialsLaguerre-Pólya classMaclaurin coefficientsRiemann ξ functionTurán inequalitiesThe 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.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SPDepartamento de matemática Aplicada IMECC UNICAMP, 13083-859 Campinas, SPDepartamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SPFAPESP: 03/01874-2FAPESP: 06/60420-0CNPq: 305622/2009-9CAPES: DGU-160Amer Mathematical SocUniversidade Estadual Paulista (Unesp)Universidade Estadual de Campinas (UNICAMP)Dimitrov, Dimitar Kolev [UNESP]Lucas, Fábio R.2014-05-27T11:25:28Z2014-05-27T11:25:28Z2011-03-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article1013-1022application/pdfhttp://dx.doi.org/10.1090/S0002-9939-2010-10515-4Proceedings of the American Mathematical Society, v. 139, n. 3, p. 1013-1022, 2011.0002-9939http://hdl.handle.net/11449/13229510.1090/S0002-9939-2010-10515-4WOS:0002887279000242-s2.0-799518462502-s2.0-79951846250.pdf1681267716971253Scopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengProceedings of the American Mathematical Society0.7071,183info:eu-repo/semantics/openAccess2023-10-27T06:06:45Zoai:repositorio.unesp.br:11449/132295Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462024-08-05T16:06:04.890283Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false |
dc.title.none.fl_str_mv |
Higher order turán inequalities for the Riemann ξ-function |
title |
Higher order turán inequalities for the Riemann ξ-function |
spellingShingle |
Higher order turán inequalities for the Riemann ξ-function Dimitrov, Dimitar Kolev [UNESP] Jensen polynomials Laguerre-Pólya class Maclaurin coefficients Riemann ξ function Turán inequalities |
title_short |
Higher order turán inequalities for the Riemann ξ-function |
title_full |
Higher order turán inequalities for the Riemann ξ-function |
title_fullStr |
Higher order turán inequalities for the Riemann ξ-function |
title_full_unstemmed |
Higher order turán inequalities for the Riemann ξ-function |
title_sort |
Higher order turán inequalities for the Riemann ξ-function |
author |
Dimitrov, Dimitar Kolev [UNESP] |
author_facet |
Dimitrov, Dimitar Kolev [UNESP] Lucas, Fábio R. |
author_role |
author |
author2 |
Lucas, Fábio R. |
author2_role |
author |
dc.contributor.none.fl_str_mv |
Universidade Estadual Paulista (Unesp) Universidade Estadual de Campinas (UNICAMP) |
dc.contributor.author.fl_str_mv |
Dimitrov, Dimitar Kolev [UNESP] Lucas, Fábio R. |
dc.subject.por.fl_str_mv |
Jensen polynomials Laguerre-Pólya class Maclaurin coefficients Riemann ξ function Turán inequalities |
topic |
Jensen polynomials Laguerre-Pólya class Maclaurin coefficients Riemann ξ function Turán inequalities |
description |
The 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. |
publishDate |
2011 |
dc.date.none.fl_str_mv |
2011-03-01 2014-05-27T11:25:28Z 2014-05-27T11:25:28Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://dx.doi.org/10.1090/S0002-9939-2010-10515-4 Proceedings of the American Mathematical Society, v. 139, n. 3, p. 1013-1022, 2011. 0002-9939 http://hdl.handle.net/11449/132295 10.1090/S0002-9939-2010-10515-4 WOS:000288727900024 2-s2.0-79951846250 2-s2.0-79951846250.pdf 1681267716971253 |
url |
http://dx.doi.org/10.1090/S0002-9939-2010-10515-4 http://hdl.handle.net/11449/132295 |
identifier_str_mv |
Proceedings of the American Mathematical Society, v. 139, n. 3, p. 1013-1022, 2011. 0002-9939 10.1090/S0002-9939-2010-10515-4 WOS:000288727900024 2-s2.0-79951846250 2-s2.0-79951846250.pdf 1681267716971253 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Proceedings of the American Mathematical Society 0.707 1,183 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
1013-1022 application/pdf |
dc.publisher.none.fl_str_mv |
Amer Mathematical Soc |
publisher.none.fl_str_mv |
Amer Mathematical Soc |
dc.source.none.fl_str_mv |
Scopus reponame:Repositório Institucional da UNESP instname:Universidade Estadual Paulista (UNESP) instacron:UNESP |
instname_str |
Universidade Estadual Paulista (UNESP) |
instacron_str |
UNESP |
institution |
UNESP |
reponame_str |
Repositório Institucional da UNESP |
collection |
Repositório Institucional da UNESP |
repository.name.fl_str_mv |
Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP) |
repository.mail.fl_str_mv |
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1808128607185272832 |