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Complete monotonicity for a ratio of finitely many gamma functions

  • Hai-Sheng Chen , Ye-Cheng Zhu EMAIL logo and Jia-Hui Wang
Published/Copyright: May 24, 2024
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Abstract

In this paper, we solve the question completely raised by Feng Qi and Dongkyu Lim in paper “Monotonicity properties for a ratio of finite many gamma functions” published in Advances in Difference Equations and get some properties about ratios of finitely many gamma functions such as complete monotonicity, logarithmically complete monotonicity, the Bernstein function property, null point and extreme value.


This work was supported by the National Natural Science Foundation of China (12026262).


  1. Communicated by Marek Balcerzak

References

[1] Abramowitz, M.—Stegun, I. A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th ed., Applied Mathematics Series, National Bureau of Standards, Dover Publications, New York, 1972.Search in Google Scholar

[2] Alzer, H.: Complete monotonicity of a function related to the binomial probability, J. Math. Anal. Appl. 459 (2018), 10–15.10.1016/j.jmaa.2017.10.077Search in Google Scholar

[3] Artin, E.: The Gamma Function, Athena Series, Holt, Rinehart and Winston, New York, 1964.Search in Google Scholar

[4] Atanassov, R.—Tsoukrovski, U.: Some properties of a class of logarithmically completely monotonic functions, C. R. Acad. Bulgare Sci. 41 (1988), 21–23.Search in Google Scholar

[5] Berg, C.: Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), 433–439.10.1007/s00009-004-0022-6Search in Google Scholar

[6] Bohr, H. A.: Laerebog i Matematisk Analyse: Laeren om de Reelle Funktioner med Anvendelse paa den Analytiske Plangeometri og Rumgeometri, Jul. Gjellerups Forlag, Copenhagen, 1922.Search in Google Scholar

[7] Besenyei, Á.: On complete monotonicity of some functions related to means, Math. Inequal. Appl. 16 (2013), 233–239.10.7153/mia-16-17Search in Google Scholar

[8] Chen, C.-P.—Qi, F.—Srivastava, H.: Some properties of functions related to the gamma and psi functions, Integral Transforms Spec. Funct. 21 (2010), 153–164.10.1080/10652460903064216Search in Google Scholar

[9] Chung, W. S.—Kim, T.—Mansour, T.: The q-deformed gamma function and q-deformed polygamma function, Bull. Korean Math. Soc. 51 (2014), 1155–1161.10.4134/BKMS.2014.51.4.1155Search in Google Scholar

[10] Guo, B.-N.—Qi, F.—Zhao, J.-L.—Luo, Q.-M.: Sharp inequalities for polygamma functions, Math. Slovaca 65 (2015), 103–120.10.1515/ms-2015-0010Search in Google Scholar

[11] Krasniqi, V.—Mansour, T.—Shabani, A. S.: Some monotonicity properties and inequalities for γ and ζ-functions, Math. Commun. 15 (2010), 365–376.Search in Google Scholar

[12] Feller, W.: Completely monotone functions and sequences, Duke Math. J. 5 (1939), 661–674.10.1215/S0012-7094-39-00555-7Search in Google Scholar

[13] Magnus, W.—Oberhettinger, F.—Soni, R.: Analytic Inequalities, Grundlehren Math. Wiss., Springer, New York, 1970.Search in Google Scholar

[14] Mitrinovic, D. S.—Pecaric, J.—Fink, A. M.: Classical and New Inequalities in Analysis, Kluwer Academic, Norwell, 1993.Search in Google Scholar

[15] Olver, F.—Lozier, D.—Boisvert, R.—Clark, C.: Nist Handbook of Mathematical Functions, Cambridge University Press, New York, 2010.Search in Google Scholar

[16] Ouimet, F.: Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex, J. Math. Anal. Appl. 466 (2018), 1609–1617.10.1016/j.jmaa.2018.06.049Search in Google Scholar

[17] Ouimet, F.—Qi, F.: Logarithmically complete monotonicity of a matrix-parametrized analogue of the multinomial distribution, Math. Inequal. Appl. 25 (2022), 703–714.10.7153/mia-2022-25-45Search in Google Scholar

[18] Qi, F.: Complete monotonicity for a new ratio of finitely many gamma functions, Acta Math. Sci. 42 (2022), 511–520.10.1007/s10473-022-0206-9Search in Google Scholar

[19] Qi, F.: A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers, J. Comput. Appl. Math. 351 (2019), 1–5.10.1016/j.cam.2018.10.049Search in Google Scholar

[20] Qi, F.—Agarwal, R. P.: On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequal. Appl. 2019 (2019), 1–42.10.1186/s13660-019-1976-zSearch in Google Scholar

[21] Qi, F.—Chapman, R. J.: Two closed forms for the Bernoulli polynomials, J. Number Theory 159 (2016), 89–100.10.1016/j.jnt.2015.07.021Search in Google Scholar

[22] Qi, F.—Guo, B.-N.: Complete monotonicity of divided differences of the di-and tri-gamma functions with applications, Georgian Math. J. 23 (2016), 279–291.10.1515/gmj-2016-0004Search in Google Scholar

[23] Qi, F.—Guo, B.-N.: From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions, J. Math. Anal. Appl. 493 (2021), Art. ID 124478.10.1016/j.jmaa.2020.124478Search in Google Scholar PubMed PubMed Central

[24] Qi, F.—Li, W.-H.: Integral representations and properties of some functions involving the logarithmic function, Filomat 30 (2016), 1659–1674.10.2298/FIL1607659QSearch in Google Scholar

[25] Qi, F.—Li, W.-H.—Yu, S.-B.—Du, X.-Y.—Guo, B.-N.: A ratio of finitely many gamma functions and its properties with applications, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115 (2021), 39–52.10.1007/s13398-020-00988-zSearch in Google Scholar

[26] Qi, F.—Lim, D.: Monotonicity properties for a ratio of finite many gamma functions, Adv. Difference Equ. 2020 (2020), Art. No. 193.10.1186/s13662-020-02655-4Search in Google Scholar PubMed PubMed Central

[27] Qi, F.—Niu, D.-W.—Lim, D.—Guo, B.-N.: Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions, Appl. Anal. Discrete Math. 14 (2020), 512–527.10.2298/AADM191111033QSearch in Google Scholar

[28] Schilling, R. L.—Song, R.—Vondracek, Z.: Bernstein Functions: Theory and Applications, de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, 2012.10.1515/9783110269338Search in Google Scholar

[29] Temme, N. M.: Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley–Interscience, New York, 1996.10.1002/9781118032572Search in Google Scholar

[30] Widder, D. V.: The Laplace Transform, Princeton University Press, Princeton, 1946.Search in Google Scholar

[31] Yang, Z.-H.—Xi, B.-Y.—Zheng, S.-Z.: Some properties of the generalized Gaussian ratio and their applications, Math. Inequal. Appl. 23 (2020), 177–200.10.7153/mia-2020-23-15Search in Google Scholar

Received: 2023-04-08
Accepted: 2023-08-10
Published Online: 2024-05-24
Published in Print: 2024-04-25

© 2024 Mathematical Institute Slovak Academy of Sciences

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