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Lower Bound of Sectional Curvature of Fisher–Rao Manifold of Beta Distributions and Complete Monotonicity of Functions Involving Polygamma Functions

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Abstract

In the paper, by virtue of convolution theorem for the Laplace transforms and analytic techniques, the author finds necessary and sufficient conditions for complete monotonicity, monotonicity, and inequalities of several functions involving polygamma functions. By these results, the author derives a lower bound of a function related to the sectional curvature of the Fisher–Rao manifold of beta distributions.

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References

  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55. Reprint of the 1972nd edn. Dover Publications Inc, New York (1992)

  2. Brigant, A.L., Puechmorel, S.: The Fisher-Rao geometry of beta distributions applied to the study of canonical moments. arXiv (2019). https://arxiv.org/abs/1904.08247

  3. Brigant, A.L., Preston, S., Puechmorel, S.: Fisher–Rao geometry of Dirichlet distributions. arXiv (2020). https://arxiv.org/abs/2005.05608

  4. Brigant, A.L., Preston, S.C., Puechmorel, S.: Fisher–Rao geometry of Dirichlet distributions. Differ. Geom. Appl. 74, 101702 (2021). https://doi.org/10.1016/j.difgeo.2020.101702

    Article  MathSciNet  MATH  Google Scholar 

  5. Gao, P.: Some monotonicity properties of gamma and \(q\)-gamma functions. ISRN Math. Anal. (2011). https://doi.org/10.5402/2011/375715

    Article  MathSciNet  MATH  Google Scholar 

  6. Guo, B.-N., Qi, F.: A completely monotonic function involving the tri-gamma function and with degree one. Appl. Math. Comput. 218(19), 9890–9897 (2012). https://doi.org/10.1016/j.amc.2012.03.075

    Article  MathSciNet  MATH  Google Scholar 

  7. Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic Publishers, Dordrecht. (1993). https://doi.org/10.1007/978-94-017-1043-5

    Book  MATH  Google Scholar 

  8. Qi, F.: Necessary and sufficient conditions for complete monotonicity and monotonicity of two functions defined by two derivatives of a function involving trigamma function. Appl. Anal. Discrete Math. (2021). https://doi.org/10.2298/AADM191111014Q

    Article  Google Scholar 

  9. Qi, F.: Complete monotonicity of a function involving the tri- and tetra-gamma functions. Proc. Jangjeon Math. Soc. 18(2), 253–264 (2015). https://doi.org/10.17777/pjms.2015.18.2.253

    Article  MathSciNet  MATH  Google Scholar 

  10. Qi, F.: Necessary and sufficient conditions for a difference constituted by four derivatives of a function involving trigamma function to be completely monotonic. Math. Inequal. Appl. 24(3), 845–855 (2021). https://doi.org/10.7153/mia-2021-24-58

    Article  MathSciNet  MATH  Google Scholar 

  11. Qi, F.: Complete monotonicity of a difference defined by four derivatives of a function containing trigamma function. OSF Preprints (2020). https://doi.org/10.31219/osf.io/56c2s

    Article  Google Scholar 

  12. Qi, F.: Decreasing monotonicity of two ratios defined by three or four polygamma functions. HAL preprint (2020). https://hal.archives-ouvertes.fr/hal-02998414

  13. Qi, F.: Lower bound of sectional curvature of manifold of beta distributions and complete monotonicity of functions involving polygamma functions. MDPI preprints (2020). https://doi.org/10.20944/preprints202011.0315.v1

  14. Qi, F.: Decreasing property and complete monotonicity of two functions constituted via three derivatives of a function involving trigamma function. Math. Slovaca 72 (2022) (in press)

  15. Qi, F.: Necessary and sufficient conditions for a ratio involving trigamma and tetragamma functions to be monotonic. Turk. J. Inequal. 5(1), 50–59 (2021)

    Google Scholar 

  16. Qi, F.: Necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic. TWMS J. Pure Appl. Math. 13(1) (2022) (in press)

  17. Qi, F.: Some properties of several functions involving polygamma functions and originating from the sectional curvature of the beta manifold. São Paulo J. Math. Sci. 14(2), 614–630 (2020). https://doi.org/10.1007/s40863-020-00193-1

    Article  MathSciNet  MATH  Google Scholar 

  18. Qi, F., Berg, C.: Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function. Mediterr. J. Math. 10(4), 1685–1696 (2013). https://doi.org/10.1007/s00009-013-0272-2

    Article  MathSciNet  MATH  Google Scholar 

  19. Qi, F., Guo, B.-N.: Complete monotonicity of divided differences of the di- and tri-gamma functions with applications. Georgian Math. J. 232, 279–291 (2016). https://doi.org/10.1515/gmj-2016-0004

    Article  MathSciNet  MATH  Google Scholar 

  20. Qi, F., Guo, B.-N.: Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic. Adv. Appl. Math. 44(1), 71–83 (2010). https://doi.org/10.1016/j.aam.2009.03.003

    Article  MathSciNet  MATH  Google Scholar 

  21. Qi, F., Han, L.-X., Yin, H.-P.: Monotonicity and complete monotonicity of two functions defined by three derivatives of a function involving trigamma function. HAL preprint (2020). https://hal.archives-ouvertes.fr/hal-02998203

  22. Schilling, R.L., Song, R., Vondraček, Z.: Bernstein Functions. de Gruyter Studies in Mathematics, vol. 37. Walter de Gruyter, Berlin (2012). https://doi.org/10.1515/9783110269338

    Book  Google Scholar 

  23. Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1946)

    Google Scholar 

  24. Yang, Z.-H.: Some properties of the divided difference of psi and polygamma functions. J. Math. Anal. Appl. 455(1), 761–777 (2017). https://doi.org/10.1016/j.jmaa.2017.05.081

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author appreciates anonymous referees for their careful corrections to, helpful suggestions to, and valuable comments on the original version of this paper.

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Authors and Affiliations

  1. Institute of Mathematics, Henan Polytechnic University, Jiaozuo, 454010, Henan, China

    Feng Qi

  2. School of Mathematical Sciences, Tiangong University, Tianjin, 300387, China

    Feng Qi

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  1. Feng Qi
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Correspondence to Feng Qi.

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Dedicated to Mr. Guo-Sheng Wang and other school teachers in my childhood.

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Qi, F. Lower Bound of Sectional Curvature of Fisher–Rao Manifold of Beta Distributions and Complete Monotonicity of Functions Involving Polygamma Functions. Results Math 76, 217 (2021). https://doi.org/10.1007/s00025-021-01530-2

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