COMPLETE MONOTONICITY FOR A NEW RATIO OF FINITELY MANY GAMMA FUNCTIONS

doi:10.1007/s10473-022-0206-9

Abstract

Abstract: In this paper, by deriving an inequality involving the generating function of the Bernoulli numbers, the author introduces a new ratio of finitely many gamma functions, finds complete monotonicity of the second logarithmic derivative of the ratio, and simply reviews the complete monotonicity of several linear combinations of finitely many digamma or trigamma functions.

Key words: Bernoulli number, ratio, generating function, complete monotonicity, gamma function, digamma function, trigamma function, logarithmic derivative, linear combination, inequality

CLC Number:

26A48

Feng QI. COMPLETE MONOTONICITY FOR A NEW RATIO OF FINITELY MANY GAMMA FUNCTIONS[J].Acta mathematica scientia,Series B, 2022, 42(2): 511-520.

Trendmd

References

[1] Abramowitz M, Stegun I A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Applied Mathematics Series 55, 10th printing. Dover New York and Washington:Publications, 1972
[2] Alzer H. Complete monotonicity of a function related to the binomial probability. J Math Anal Appl, 2018, 459(1):10-15. https://doi.org/10.1016/j.jmaa.2017.10.077
[3] Alzer H, Berg C. Some classes of completely monotonic functions. II. Ramanujan J, 2006, 11(2):225-248. https://doi.org/10.1007/s11139-006-6510-5
[4] Bullen P S. Handbook of Means and Their Inequalities. Mathematics and its Applications 560. Dordrecht:Kluwer Academic Publishers Group, 2003. https://doi.org/10.1007/978-94-017-0399-4
[5] Cringanu J. Inequalities associated with ratios of gamma functions. Bull Aust Math Soc, 2018, 97(3):453-458. https://doi.org/10.1017/S0004972718000138
[6] Guo B-N, Qi F. On complete monotonicity of linear combination of finite psi functions. Commun Korean Math Soc, 2019, 34(4):1223-1228. https://doi.org/10.4134/CKMS.c180430
[7] Guo B-N, Qi F. Properties and applications of a function involving exponential functions. Commun Pure Appl Anal, 2009, 8(4):1231-1249. https://doi.org/10.3934/cpaa.2009.8.1231
[8] Guo B-N, Qi F. Two new proofs of the complete monotonicity of a function involving the psi function. Bull Korean Math Soc, 2010, 47(1):103-111. https://doi.org/10.4134/bkms.2010.47.1.103
[9] Guo B-N, Qi F, Zhao J-L, Luo Q-M. Sharp inequalities for polygamma functions. Math Slovaca, 2015, 65(1):103-120 https://doi.org/10.1515/ms-2015-0010.
[10] Gurland J. An inequality satisfied by the gamma function. Skand Aktuarietidskr, 1956, 39:171-172. https://doi.org/10.1080/03461238.1956.10414947
[11] Leblanc A, Johnson B C. A Family of Inequalities Related to Binomial Probabilities. Department of Statistics, University of Manitoba. Tech Report, 2006-03
[12] Leblanc A, Johnson B C. On a uniformly integrable family of polynomials defined on the unit interval. J Inequal Pure Appl Math, 2007, 8(3):Article 67, 5 pages. https://www.emis.de/journals/JIPAM/article878.html
[13] Lü Y-P, Sun T-C, Chu Y-M. Necessary and sufficient conditions for a class of functions and their reciprocals to be logarithmically completely monotonic. J Inequal Appl, 2011(Paper No 36):8 pages. https://doi.org/10.1186/1029-242X-2011-36
[14] Mitrinović D S. Analytic Inequalities. New York-Berlin:Springer-Verlag, 1970, Band 165
[15] Mitrinović D S, Pečarić J E, Fink A M. Classical and New Inequalities in Analysis. Kluwer Academic Publishers, 1993. https://doi.org/10.1007/978-94-017-1043-5
[16] Olver F W J, Lozier D W, Boisvert R F, Clark C W. NIST Handbook of Mathematical Functions. New York:Cambridge University Press, 2010. http://dlmf.nist.gov/
[17] Ouimet F. Complete monotonicity of a ratio of gamma functions and some combinatorial inequalities for multinomial coefficients. arXiv, 2019. https://arxiv.org/abs/1907.05262
[18] Ouimet F. Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex. J Math Anal Appl, 2018, 466(2):1609-1617. https://doi.org/10.1016/j.jmaa.2018.06.049
[19] Qi F. A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers. J Comput Appl Math, 2019, 351:1-5. https://doi.org/10.1016/j.cam.2018.10.049
[20] Qi F. A logarithmically completely monotonic function involving the q-gamma function. J Nonlinear Convex Anal, 2022. https://hal.archives-ouvertes.fr/hal-01803352v1
[21] Qi F. Bounds for the ratio of two gamma functions. J Inequal Appl, 2010, 2010(Article ID 493058):84 pages. https://doi.org/10.1155/2010/493058
[22] Qi F. Bounds for the ratio of two gamma functions:from Gautschi's and Kershaw's inequalities to complete monotonicity. Turkish J Anal Number Theory, 2014, 2(5):152-164. https://doi.org/10.12691/tjant-2-5-1
[23] Qi F. Complete monotonicity for a new ratio of finite many gamma functions. HAL, 2020. https://hal.archives-ouvertes.fr/hal-02511909v1
[24] Qi F. Complete monotonicity of functions involving the q-trigamma and q-tetragamma functions. Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM, 2015, 109(2):419-429. https://doi.org/10.1007/s13398- 014-0193-3
[25] Qi F. Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities. Filomat, 2013, 27(4):601-604. https://doi.org/10.2298/FIL1304601Q
[26] Qi F. Notes on a double inequality for ratios of any two neighbouring non-zero Bernoulli numbers. Turkish J Anal Number Theory, 2018, 6(5):129-131. https://doi.org/10.12691/tjant-6-5-1
[27] Qi F, Agarwal R P. On complete monotonicity for several classes of functions related to ratios of gamma functions. J Inequal Appl, 2019, 2019(Paper No 36):42 pages. https://doi.org/10.1186/s13660-019-1976-z
[28] Qi F, Chapman R J. Two closed forms for the Bernoulli polynomials. J Number Theory, 2016, 159:89-100. https://doi.org/10.1016/j.jnt.2015.07.021
[29] Qi F, Guo B-N. Complete monotonicity of divided differences of the di- and tri-gamma functions with applications. Georgian Math J, 2016, 23(2):279-291. https://doi.org/10.1515/gmj-2016-0004
[30] 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, 2021, 493(1):Article 124478, 19 pages. https://doi.org/10.1016/j.jmaa.2020.124478
[31] Qi F, Guo B-N. Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function. Rev R Acad Cienc Exactas Fís Nat Ser A Math RACSAM, 2017, 111(2):425-434. https://doi.org/10.1007/s13398-016-0302-6
[32] Qi F, Guo B-N, Debnath L. A lower bound for ratio of power means. Int J Math Math Sci, 2004, 2014(1/4):49-53. https://doi.org/10.1155/S0161171204208158
[33] Qi F, Li W -H. Integral representations and properties of some functions involving the logarithmic function. Filomat, 2016, 30(7):1659-1674. https://doi.org/10.2298/FIL1607659Q
[34] 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 Math RACSAM, 2021, 115(2):Paper No 39, 14 pages. https://doi.org/10.1007/s13398-020-00988-z
[35] Qi F, Lim D. Monotonicity properties for a ratio of finite many gamma functions. Adv Difference Equ, 2020, 2020(Paper No 193):9 pages. https://doi.org/10.1186/s13662-020-02655-4
[36] Qi F, Liu A-Q. Completely monotonic degrees for a difference between the logarithmic and psi functions. J Comput Appl Math, 2019, 361:366-371. https://doi.org/10.1016/j.cam.2019.05.001
[37] Qi F, Luo Q-M. Bounds for the ratio of two gamma functions-From Wendel's and related inequalities to logarithmically completely monotonic functions. Banach J Math Anal, 2012, 6(2):132-158. https://doi.org/10.15352/bjma/1342210165
[38] Qi F, Luo Q-M. Bounds for the ratio of two gamma functions:from Wendel's asymptotic relation to Elezović-Giordano-Pečarić's theorem. J Inequal Appl, 2013, 2013(Paper No 542):20 pages. https://doi.org/10.1186/1029-242X-2013-542
[39] Qi F, Mei J-Q, Xia D-F, Xu S-L. New proofs of weighted power mean inequalities and monotonicity for generalized weighted mean values. Math Inequal Appl, 2000, 3(3):377-383. https://doi.org/10.7153/mia- 03-38
[40] 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, 2020, 14(2):512-527. https://doi.org/10.2298/AADM191111033Q
[41] Schilling R L, Song R, Vondraček Z. Bernstein Functions. 2nd ed. de Gruyter Studies in Mathematics 37. Berlin, Germany:Walter de Gruyter, 2012. https://doi.org/10.1515/9783110269338
[42] Salem A, Kamel E S. Some completely monotonic functions associated with the q-gamma and the q-polygamma functions. Acta Math Sci, 2015, 35B(5):1214-1224. https://doi.org/10.1016/S0252- 9602(15)30050-3
[43] Shen J-M, Yang Z-H, Qian W-M, Zhang W, Chu Y-M. Sharp rational bounds for the gamma function. Math Inequal Appl, 2020, 23(3):843-853. https://doi.org/10.7153/mia-2020-23-68
[44] Shuang Y, Guo B-N, Qi F. Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios. Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM, 2021, 115(3):Paper No 135, 12 pages. https://doi.org/10.1007/s13398-021-01071-x
[45] Temme N M. Special Functions:An Introduction to Classical Functions of Mathematical Physics, A WileyInterscience Publication. New York:John Wiley & Sons, Inc, 1996. https://doi.org/10.1002/9781118032572
[46] Tian J-F, Yang Z-H. Asymptotic expansions of Gurland's ratio and sharp bounds for their remainders. J Math Anal Appl, 2021, 493(2):Paper No 124545, 19 pp. https://doi.org/10.1016/j.jmaa.2020.124545
[47] Widder D V. The Laplace Transform. Princeton:Princeton University Press, 1946
[48] Yang Z-H, Qian W-M, Chu Y-M, Zhang W. On rational bounds for the gamma function, J Inequal Appl, 2017, 2017(Paper No 210):17 pages. https://doi.org/10.1186/s13660-017-1484-y
[49] Yang Z-H, Tian J-F. A class of completely mixed monotonic functions involving the gamma function with applications. Proc Amer Math Soc, 2018, 146(11):4707-4721. https://doi.org/10.1090/proc/14199
[50] Yang Z-H, Tian J-F. Sharp bounds for the ratio of two zeta functions. J Comput Appl Math, 2020, 364(Paper No 112359):14 pages. https://doi.org/10.1016/j.cam.2019.112359
[51] Yang Z-H, Xi B-Y, Zheng S-Z. Some properties of the generalized Gaussian ratio and their applications. Math Inequal Appl, 2020, 23(1):177-200. https://doi.org/10.7153/mia-2020-23-15
[52] Yang Z-H, Zhang W, Chu Y-M. Sharp Gautschi inequality for parameter 0< p < 1 with applications. Math Inequal Appl, 2017, 20(4):1107-1120. https://doi.org/10.7153/mia-2017-20-71
[53] Yang Z-H, Zheng S-Z. Complete monotonicity and inequalites involving Gurland's ratios of gamma functions. Math Inequal Appl, 2019, 22(1):97-109. https://doi.org/10.7153/mia-2019-22-07
[54] Yin L, Huang L-G. Limit formulas related to the p-gamma and p-polygamma functions at their singularities. Filomat, 2015, 29(7):1501-1505. https://doi.org/10.2298/FIL1507501Y
[55] Zhao T-H, Chu Y-M, Wang H. Logarithmically complete monotonicity properties relating to the gamma function. Abstr Appl Anal, 2011, 2011(Art ID 896483):13 pages. https://doi.org/10.1155/2011/896483
[56] Zhu L. New bounds for the ratio of two adjacent even-indexed Bernoulli numbers. Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM, 2020, 114(2):Paper No 83, 13 pages. https://doi.org/10.1007/s13398-020- 00814-6