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Decreasing properties of two ratios defined by three and four polygamma functions
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 89-101.

In the paper, by virtue of the convolution theorem for the Laplace transforms, with the aid of three monotonicity rules for the ratios of two functions, of two definite integrals, and of two Laplace transforms, in terms of the majorization, and in the light of other analytic techniques, the author presents decreasing properties of two ratios defined by three and four polygamma functions.

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DOI : 10.5802/crmath.296
Classification : 33B15, 26A48, 26A51, 26D07, 26D15, 26D20, 44A10, 60E15

Feng Qi 1

1 Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454003, Henan, China; School of Mathematical Sciences, Tiangong University, Tianjin 300387, China
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Decreasing properties of two ratios defined by three and four polygamma functions},
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Feng Qi. Decreasing properties of two ratios defined by three and four polygamma functions. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 89-101. doi : 10.5802/crmath.296. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.296/

[1] Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Milton Abramowitz; Irene A. Stegun, eds.), National Bureau of Standards, Applied Mathematics Series, 55, Dover Publications, 1992 (reprint of the 1972 edition)

[2] Horst Alzer; Jim Wells Inequalities for the polygamma functions, SIAM J. Math. Anal., Volume 29 (1998) no. 6, pp. 1459-1466 | DOI | MR | Zbl

[3] Glen Douglas Anderson; Mavina Krishna Vamanamurthy; Matti Vuorinen Conformal Invariants, Inequalities, and Quasiconformal Maps, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, 1997 | Zbl

[4] Necdet Batir On some properties of digamma and polygamma functions, J. Math. Anal. Appl., Volume 328 (2007) no. 1, pp. 452-465 | DOI | MR | Zbl

[5] Yuming Chu; Xiaoming Zhang Necessary and sufficient conditions such that extended mean values are Schur-convex or Schur-concave, J. Math. Kyoto Univ., Volume 48 (2008) no. 1, pp. 229-238 | MR | Zbl

[6] Peng Gao Some monotonicity properties of gamma and q-gamma functions, ISRN Math. Anal., Volume 2011 (2011), 375715, 15 pages | MR | Zbl

[7] Peng Gao Some completely monotonic functions involving the polygamma functions, J. Inequal. Appl., Volume 2019 (2019), 218, 9 pages | MR | Zbl

[8] Bai-Ni Guo; Feng Qi On the increasing monotonicity of a sequence originating from computation of the probability of intersecting between a plane couple and a convex body, Turk. J. Anal. Number Theory, Volume 3 (2015) no. 1, pp. 21-23

[9] Bai-Ni Guo; Feng Qi; Hari M. Srivastava Some uniqueness results for the non-trivially complete monotonicity of a class of functions involving the polygamma and related functions, Integral Transforms Spec. Funct., Volume 21 (2010) no. 11, pp. 849-858 | MR

[10] John Gurland An inequality satisfied by the gamma function, Skand. Aktuarietidskr., Volume 39 (1956), pp. 171-172 | MR | Zbl

[11] Alice Le Brigant; Stephen C. Preston; Stéphane Puechmorel Fisher–Rao geometry of Dirichlet distributions, Differ. Geom. Appl., Volume 74 (2021), 101702, 17 pages | MR | Zbl

[12] Albert W. Marshall; Ingram Olkin; Barry C. Arnold Inequalities: Theory of Majorization and its Applications, Springer Series in Statistics, Springer, 2011

[13] Dragoslav S. Mitrinović; Josip E. Pečarić; Aarlington M. Fink Classical and New Inequalities in Analysis, Mathematics and Its Applications. East European Series, 61, Kluwer Academic Publishers, 1993 | DOI

[14] Feng Qi Completely monotonic degree of a function involving trigamma and tetragamma functions, AIMS Math., Volume 5 (2020) no. 4, pp. 3391-3407 | MR

[15] Feng Qi Decreasing monotonicity of two ratios defined by three or four polygamma functions (2020) (https://hal.archives-ouvertes.fr/hal-02998414)

[16] Feng Qi Lower bound of sectional curvature of manifold of beta distributions and complete monotonicity of functions involving polygamma functions (2020) (https://www.preprints.org/manuscript/202011.0315/v1) | DOI

[17] Feng Qi Necessary and sufficient conditions for a difference defined by four derivatives of a function containing trigamma function to be completely monotonic (2020) (to appear in Appl. Comput. Math., https://osf.io/56c2s/) | DOI

[18] Feng Qi Some properties of several functions involving polygamma functions and originating from the sectional curvature of the beta manifold, São Paulo J. Math. Sci., Volume 14 (2020) no. 2, pp. 614-630 | MR | Zbl

[19] Feng Qi Bounds for completely monotonic degree of a remainder for an asymptotic expansion of the trigamma function, Arab. J. Basic Appl. Sci., Volume 28 (2021) no. 1, pp. 314-318

[20] Feng Qi Lower bound of sectional curvature of Fisher–Rao manifold of beta distributions and complete monotonicity of functions involving polygamma functions, Results Math., Volume 76 (2021) no. 4, 217, 16 pages | MR | Zbl

[21] Feng Qi Necessary and sufficient conditions for a difference constituted by four derivatives of a function involving trigamma function to be completely monotonic, Math. Inequal. Appl., Volume 24 (2021) no. 3, pp. 845-855 | MR | Zbl

[22] Feng Qi Necessary and sufficient conditions for a ratio involving trigamma and tetragamma functions to be monotonic, Turk. J. Inequal., Volume 5 (2021) no. 1, pp. 50-59

[23] Feng Qi 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., Volume 15 (2021) no. 2, pp. 378-392 | DOI | MR

[24] Feng Qi Decreasing property and complete monotonicity of two functions constituted via three derivatives of a function involving trigamma function, Math. Slovaca, Volume 72 (2022) (in press)

[25] Feng Qi Two completely monotonic functions defined by two derivatives of a function involving trigamma function, TWMS J. Pure Appl. Math., Volume 13 (2022) no. 1 (in press)

[26] Feng Qi; R. P. Agarwal On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequal. Appl., Volume 2019 (2019), 36, 42 pages | MR | Zbl

[27] Feng Qi; Bai-Ni Guo Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic, Adv. Appl. Math., Volume 44 (2010) no. 1, pp. 71-83 | MR | Zbl

[28] Feng Qi; Bai-Ni Guo Complete monotonicity of divided differences of the di- and tri-gamma functions with applications, Georgian Math. J., Volume 23 (2016) no. 2, pp. 279-291 | MR | Zbl

[29] Feng Qi; Ling-Xiong Han; Hong-Ping Yin Monotonicity and complete monotonicity of two functions defined by three derivatives of a function involving trigamma function (2020) (https://hal.archives-ouvertes.fr/hal-02998203)

[30] Feng Qi; Wen-Hui Li; Shu-Bin Yu; Xin-Yu Du; Bai-Ni Guo A ratio of finitely many gamma functions and its properties with applications, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM, Volume 115 (2021) no. 2, 39, 14 pages | MR | Zbl

[31] Feng Qi; Ai-Qi Liu Completely monotonic degrees for a difference between the logarithmic and psi functions, J. Comput. Appl. Math., Volume 361 (2019), pp. 366-371 | MR | Zbl

[32] Feng Qi; Qiu-Ming Luo Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezović-Giordano-Pečarić’s theorem, J. Inequal. Appl., Volume 2013 (2013), 542, 20 pages | Zbl

[33] Feng Qi; Mansour Mahmoud Some properties of a function originating from geometric probability for pairs of hyperplanes intersecting with a convex body, Math. Comput. Appl., Volume 21 (2016) no. 3, 27, 6 pages | MR

[34] Feng Qi; Cristinel Mortici; Bai-Ni Guo Some properties of a sequence arising from geometric probability for pairs of hyperplanes intersecting with a convex body, Comput. Appl. Math., Volume 37 (2018) no. 2, pp. 2190-2200 | MR | Zbl

[35] Feng Qi; Xiao-Ting Shi; Mansour Mahmoud; Fang-Fang Liu Schur-convexity of the Catalan–Qi function related to the Catalan numbers, Tbil. Math. J., Volume 9 (2016) no. 2, pp. 141-150 | MR | Zbl

[36] René L. Schilling; Renming Song; Zoran Vondraček Bernstein Functions, De Gruyter Studies in Mathematics, 37, Walter de Gruyter, 2012 | DOI

[37] H.-N. Shi Two Schur-convex functions related to Hadamard-type integral inequalities, Publ. Math., Volume 78 (2011) no. 2, pp. 393-403 | MR | Zbl

[38] Jing-Feng Tian; Zhen-Hang Yang Asymptotic expansions of Gurland’s ratio and sharp bounds for their remainders, J. Math. Anal. Appl., Volume 493 (2021) no. 2, 124545, 19 pages | MR | Zbl

[39] Jing-Feng Tian; Zhen-Hang Yang New properties of the divided difference of psi and polygamma functions, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM, Volume 115 (2021) no. 3, 147, 23 pages | MR | Zbl

[40] David V. Widder The Laplace Transform, Princeton University Press, 1946

[41] Ai-Min Xu; Zhong-Di Cen Qi’s conjectures on completely monotonic degrees of remainders of asymptotic formulas of di- and tri-gamma functions, J. Inequal. Appl., Volume 2020 (2020), 83, 10 pages | Zbl

[42] Zhen-Hang Yang Some properties of the divided difference of psi and polygamma functions, J. Math. Anal. Appl., Volume 455 (2017) no. 1, pp. 761-777 | DOI | MR | Zbl

[43] Zhen-Hang Yang; Jing-Feng Tian Monotonicity and inequalities for the gamma function, J. Inequal. Appl., Volume 2017 (2017), 317, 15 pages | MR | Zbl

[44] Zhen-Hang Yang; Jing-Feng Tian A class of completely mixed monotonic functions involving the gamma function with applications, Proc. Am. Math. Soc., Volume 146 (2018) no. 11, pp. 4707-4721 | DOI | MR | Zbl

[45] Zhen-Hang Yang; Jing-Feng Tian Monotonicity rules for the ratio of two Laplace transforms with applications, J. Math. Anal. Appl., Volume 470 (2019) no. 2, pp. 821-845 | DOI | MR | Zbl

[46] Zhen-Hang Yang; Bo-Yan Xi; Shen-Zhou Zheng Some properties of the generalized Gaussian ratio and their applications, Math. Inequal. Appl., Volume 23 (2020) no. 1, pp. 177-200 | MR | Zbl

[47] Hong-Ping Yin; Xi-Min Liu; Jing-Yu Wang; Bai-Ni Guo Necessary and sufficient conditions on the Schur convexity of a bivariate mean, AIMS Math., Volume 6 (2021) no. 1, pp. 296-303 | MR

[48] Jiang-Fu Zhao; Peng Xie; Jun Jiang Geometric probability for pairs of hyperplanes intersecting with a convex body, Math. Appl., Volume 29 (2016) no. 1, pp. 233-238 | MR | Zbl

[49] Ling Zhu Completely monotonic integer degrees for a class of special functions, AIMS Math., Volume 5 (2020) no. 4, pp. 3456-3471 | MR

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