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Turán Type Inequalities for Tricomi Confluent Hypergeometric Functions

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Abstract

Some sharp two-sided Turán type inequalities for parabolic cylinder functions and Tricomi confluent hypergeometric functions are deduced. The proofs are based on integral representations for quotients of parabolic cylinder functions and Tricomi confluent hypergeometric functions, which arise in the study of the infinite divisibility of the Fisher–Snedecor F distribution. Moreover, some complete monotonicity results are given concerning Turán determinants of Tricomi confluent hypergeometric functions. These complement and improve some of the results of Ismail and Laforgia (in Constr. Approx. 26:1–9, 2007).

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Acknowledgements

The research of Á. Baricz was supported in part by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and in part by the Romanian National Council for Scientific Research in Education CNCSIS-UEFISCSU, project number PN-II-RU-PD 388/2012, and was completed during his visit in September 2011 to City University of Hong Kong. This author is grateful to the Department of Mathematics of City University of Hong Kong for hospitality. The research of M.E.H. Ismail was partially supported by the NPST Program of King Saud University, Riyadh, project number 10-MAT 1293-02 and by the Research Grants Council of Hong Kong under contract # 101411. This work was carried out while M.E.H. Ismail was affiliated with the Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong. Both of the authors are grateful to the referees for extensive comments and constructive criticisms that improved the presentation of the results.

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Authors and Affiliations

  1. Department of Economics, Babeş-Bolyai University, Cluj-Napoca, 400591, Romania

    Árpád Baricz

  2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Mourad E. H. Ismail

  3. Department of Mathematics, King Saud University, Riyadh, Saudi Arabia

    Mourad E. H. Ismail

Authors
  1. Árpád Baricz
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  2. Mourad E. H. Ismail
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Correspondence to Árpád Baricz.

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Communicated by Edward B. Saff.

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Baricz, Á., Ismail, M.E.H. Turán Type Inequalities for Tricomi Confluent Hypergeometric Functions. Constr Approx 37, 195–221 (2013). https://doi.org/10.1007/s00365-012-9171-1

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  • DOI: https://doi.org/10.1007/s00365-012-9171-1

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