Research article

Completely monotonic degree of a function involving trigamma and tetragamma functions

  • Received: 19 November 2019 Accepted: 25 March 2020 Published: 01 April 2020
  • MSC : Primary: 33B15; Secondary: 26A12, 26A48, 26A51, 42A85, 44A10, 44A35

  • Let $\psi(x)$ be the digamma function. In the paper, the author reviews backgrounds and motivations to compute complete monotonic degree of the function $\Psi(x) = [\psi'(x)]^2+\psi''(x)$ with respect to $x\in(0, \infty)$, confirms that completely monotonic degree of the function $\Psi(x)$ is $4$, finds a relation between strongly completely monotonic functions and completely monotonic degrees, provides a proof for the relation between strongly completely monotonic functions and completely monotonic degrees, proves a property of logarithmically concave functions, and poses two open problems on lower bound for convolution of logarithmically concave functions and on completely monotonic degree of a function involving $\Psi(x)$.

    Citation: Feng Qi. Completely monotonic degree of a function involving trigamma and tetragamma functions[J]. AIMS Mathematics, 2020, 5(4): 3391-3407. doi: 10.3934/math.2020219

    Related Papers:

  • Let $\psi(x)$ be the digamma function. In the paper, the author reviews backgrounds and motivations to compute complete monotonic degree of the function $\Psi(x) = [\psi'(x)]^2+\psi''(x)$ with respect to $x\in(0, \infty)$, confirms that completely monotonic degree of the function $\Psi(x)$ is $4$, finds a relation between strongly completely monotonic functions and completely monotonic degrees, provides a proof for the relation between strongly completely monotonic functions and completely monotonic degrees, proves a property of logarithmically concave functions, and poses two open problems on lower bound for convolution of logarithmically concave functions and on completely monotonic degree of a function involving $\Psi(x)$.


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    [1] D. S. Mitrinović, J. E. Pečarić, A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993.
    [2] R. L. Schilling, R. Song, Z. Vondraček, Bernstein Functions-Theory and Applications, 2Eds., de Gruyter Studies in Mathematics 37, Walter de Gruyter, Berlin, Germany, 2012.
    [3] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946.
    [4] B. N. Guo, F. Qi, A completely monotonic function involving the tri-gamma function and with degree one, Appl. Math. Comput., 218 (2012), 9890-9897.
    [5] B. N. Guo, F. Qi, On the degree of the weighted geometric mean as a complete Bernstein function, Afr. Mat., 26 (2015), 1253-1262. doi: 10.1007/s13370-014-0279-2
    [6] F. Qi, Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions, Math. Inequal. Appl., 18 (2015), 493-518.
    [7] F. Qi, A. Q. Liu, Completely monotonic degrees for a difference between the logarithmic and psi functions, J. Comput. Appl. Math., 361 (2019), 366-371. doi: 10.1016/j.cam.2019.05.001
    [8] F. Qi, S. H. Wang, Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions, Glob. J. Math. Anal., 2 (2014), 91-97.
    [9] F. Qi, X. J. Zhang, W. H. Li, The harmonic and geometric means are Bernstein functions, Bol. Soc. Mat. Mex., 23 (2017), 713-736. doi: 10.1007/s40590-016-0085-y
    [10] F. Qi, Completely monotonic degree of remainder of asymptotic expansion of trigamma function, arXiv preprint, 2020, Available from: https://arxiv.org/abs/2003.05300v1.
    [11] F. Qi, R. P. Agarwal, On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequal. Appl., 36 (2019), 42.
    [12] F. Qi, W. H. Li, Integral representations and properties of some functions involving the logarithmic function, Filomat, 30 (2016), 1659-1674. doi: 10.2298/FIL1607659Q
    [13] F. Qi, M. Mahmoud, Completely monotonic degrees of remainders of asymptotic expansions of the digamma function, HAL preprint, 2019, Available from: https://hal.archives-ouvertes.fr/hal-02415224v1.
    [14] S. Koumandos, Monotonicity of some functions involving the gamma and psi functions, Math. Comput., 77 (2008), 2261-2275. doi: 10.1090/S0025-5718-08-02140-6
    [15] S. Koumandos, M. Lamprecht, Complete monotonicity and related properties of some special functions, Math. Comput., 82 (2013), 282, 1097-1120.
    [16] S. Koumandos, M. Lamprecht, Some completely monotonic functions of positive order, Math. Comput., 79 (2010), 1697-1707. doi: 10.1090/S0025-5718-09-02313-8
    [17] S. Koumandos, H. L. Pedersen, Absolutely monotonic functions related to Euler's gamma function and Barnes' double and triple gamma function, Monatsh. Math., 163 (2011), 51-69. doi: 10.1007/s00605-010-0197-9
    [18] S. Koumandos, H. L. Pedersen, Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function, J. Math. Anal. Appl., 355 (2009), 33-40. doi: 10.1016/j.jmaa.2009.01.042
    [19] F. Qi, B. N. Guo, Lévy-Khintchine representation of Toader-Qi mean, Math. Inequal. Appl., 21 (2018), 421-431.
    [20] F. Qi, B. N. Guo, The reciprocal of the weighted geometric mean of many positive numbers is a Stieltjes function, Quaest. Math., 41 (2018), 653-664. doi: 10.2989/16073606.2017.1396508
    [21] F. Qi, D. Lim, Integral representations of bivariate complex geometric mean and their applications, J. Comput. Appl. Math., 330 (2018), 41-58. doi: 10.1016/j.cam.2017.08.005
    [22] F. Qi, X. J. Zhang, W. H. Li, Lévy-Khintchine representations of the weighted geometric mean and the logarithmic mean, Mediterr. J. Math., 11 (2014), 315-327. doi: 10.1007/s00009-013-0311-z
    [23] B. N. Guo, F. Qi, On complete monotonicity of linear combination of finite psi functions, Commun. Korean Math. Soc., 34 (2019), 1223-1228.
    [24] F. Qi, P. Cerone, Some properties of the Fuss-Catalan numbers, Mathematics, 6 (2018), 12.
    [25] F. Qi, X. T. Shi, P. Cerone, A unified generalization of the Catalan, Fuss, and Fuss-Catalan numbers, Math. Comput. Appl., 24 (2019), 16.
    [26] Z. H. Yang, J. F. Tian, A class of completely mixed monotonic functions involving the gamma function with applications, Proc. Amer. Math. Soc., 146 (2018), 4707-4721. doi: 10.1090/proc/14199
    [27] Z. H. Yang, J. F. Tian, M. H. Ha, A new asymptotic expansion of a ratio of two gamma functions and complete monotonicity for its remainder, Proc. Amer. Math. Soc., 148 (2020), 2163-2178. doi: 10.1090/proc/14917
    [28] Z. H. Yang, J. F. Tian, Sharp bounds for the ratio of two zeta functions, J. Comput. Appl. Math., 364 (2020), 112359, 14.
    [29] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Washington, 1972.
    [30] H. Alzer, Sharp inequalities for the digamma and polygamma functions, Forum Math., 16 (2004), 181-221. doi: 10.1515/form.2004.009
    [31] N. Batir, An interesting double inequality for Euler's gamma function, J. Inequal. Pure Appl. Math., 5 (2004), 97, Available from: http://www.emis.de/journals/JIPAM/article452.html.
    [32] N. Batir, Some new inequalities for gamma and polygamma functions, J. Inequal. Pure Appl. Math., 6 (2005), 103, Available from: http://www.emis.de/journals/JIPAM/article577.html.
    [33] H. Alzer, A. Z. Grinshpan, Inequalities for the gamma and q-gamma functions, J. Approx. Theory, 144 (2007), 67-83. doi: 10.1016/j.jat.2006.04.008
    [34] B. N. Guo, F. Qi, Sharp inequalities for the psi function and harmonic numbers, Analysis (Berlin) 34 (2014), 201-208.
    [35] F. Qi, Complete monotonicity of functions involving the q-trigamma and q-tetragamma functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM., 109 (2015), 419-429. doi: 10.1007/s13398-014-0193-3
    [36] F. Qi, B. N. Guo, Necessary and sufficient conditions for functions involving the tri- and tetragamma functions to be completely monotonic, Adv. Appl. Math., 44 (2010), 71-83. doi: 10.1016/j.aam.2009.03.003
    [37] N. Batir, On some properties of digamma and polygamma functions, J. Math. Anal. Appl., 328 (2007), 452-465. doi: 10.1016/j.jmaa.2006.05.065
    [38] F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl., 2010 (2010), Article ID 493058, 84.
    [39] F. Qi, Bounds for the ratio of two gamma functions: from Gautschi's and Kershaw's inequalities to complete monotonicity, Turkish J. Anal. Number Theory, 2 (2014), 152-164. doi: 10.12691/tjant-2-5-1
    [40] F. Qi, Q. M. Luo, Bounds for the ratio of two gamma functions-From Wendel's and related inequalities to logarithmically completely monotonic functions, Banach J. Math. Anal., 6 (2012), 132-158. doi: 10.15352/bjma/1342210165
    [41] F. Qi, Q. M. Luo, Bounds for the ratio of two gamma functions: from Wendel's asymptotic relation to Elezović-Giordano-Pečarić's theorem, J. Inequal. Appl., 2013 (2013): 20.
    [42] B. N. Guo, F. Qi, H. 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., 21 (2010), 849-858. doi: 10.1080/10652461003748112
    [43] B. N. Guo, J. L. Zhao, F. Qi, A completely monotonic function involving the tri- and tetra-gamma functions, Math. Slovaca, 63 (2013), 469-478. doi: 10.2478/s12175-013-0109-2
    [44] J. L. Zhao, B. N. Guo, F. Qi, Complete monotonicity of two functions involving the tri- and tetragamma functions, Period. Math. Hungar., 65 (2012), 147-155. doi: 10.1007/s10998-012-9562-x
    [45] F. Qi, Complete monotonicity of a function involving the tri- and tetra-gamma functions, Proc. Jangjeon Math. Soc., 18 (2015), 253-264.
    [46] D. K. Kazarinoff, On Wallis' formula, Edinburgh Math. Notes, 1956 (1956), 19-21. doi: 10.1017/S095018430000029X
    [47] B. N. Guo, F. Qi, A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications, J. Korean Math. Soc., 48 (2011), 655-667. doi: 10.4134/JKMS.2011.48.3.655
    [48] F. Qi, P. Cerone, S. S. Dragomir, Complete monotonicity of a function involving the divided difference of psi functions, Bull. Aust. Math. Soc., 88 (2013), 309-319. doi: 10.1017/S0004972712001025
    [49] F. Qi, B. N. Guo, Complete monotonicity of divided differences of the di- and tri-gamma functions with applications, Georgian Math. J., 23 (2016), 279-291.
    [50] F. Qi, B. N. Guo, Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications, Commun. Pure Appl. Anal., 8 (2009), 1975-1989. doi: 10.3934/cpaa.2009.8.1975
    [51] F. Qi, Q. M. Luo, B. N. Guo, Complete monotonicity of a function involving the divided difference of digamma functions, Sci. China Math., 56 (2013), 2315-2325. doi: 10.1007/s11425-012-4562-0
    [52] F. Qi, W. H. Li, A logarithmically completely monotonic function involving the ratio of gamma functions, J. Appl. Anal. Comput., 5 (2015), 626-634.
    [53] F. Qi, L. Debnath, Evaluation of a class of definite integrals, Internat. J. Math. Ed. Sci. Tech., 32 (2001), 629-633. doi: 10.1080/00207390116734
    [54] P. R. Beesack, Inequalities involving iterated kernels and convolutions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No., 274 (1969), 11-16.
    [55] C. O. Imoru, A remark on inequalities involving convolutions, J. Math. Anal. Appl., 164 (1992), 325-336. doi: 10.1016/0022-247X(92)90117-V
    [56] D. S. Mitrinović, Analytic inequalities, In cooperation with P. M. Vasić, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970.
    [57] F. Qi, Integral representations and complete monotonicity related to the remainder of Burnside's formula for the gamma function, J. Comput. Appl. Math., 268 (2014), 155-167. doi: 10.1016/j.cam.2014.03.004
    [58] F. Qi, Absolute monotonicity of a function involving the exponential function, Glob. J. Math. Anal., 2 (2014), 184-203.
    [59] S. Y. Trimble, J. Wells, F. T. Wright, Superadditive functions and a statistical application, SIAM J. Math. Anal., 20 (1989), 1255-1259. doi: 10.1137/0520082
    [60] B. N. Guo, F. Qi, A simple proof of logarithmic convexity of extended mean values, Numer. Algorithms, 52 (2009), 89-92. doi: 10.1007/s11075-008-9259-7
    [61] B. N. Guo, F. Qi, Generalization of Bernoulli polynomials, Int. J. Math. Ed. Sci. Tech., 33 (2002), 428-431. doi: 10.1080/002073902760047913
    [62] B. N. Guo, F. Qi, Properties and applications of a function involving exponential functions, Commun. Pure Appl. Anal., 8 (2009), 1231-1249. doi: 10.3934/cpaa.2009.8.1231
    [63] B. N. Guo, F. Qi, The function (bx - ax)/x: Logarithmic convexity and applications to extended mean values, Filomat, 25 (2011), 63-73. doi: 10.2298/FIL1104063G
    [64] S. Guo, F. Qi, A class of completely monotonic functions related to the remainder of Binet's formula with applications, Tamsui Oxf. J. Math. Sci., 25 (2009), 9-14.
    [65] M. Masjed-Jamei, F. Qi, H. M. Srivastava, Generalizations of some classical inequalities via a special functional property, Integral Transforms Spec. Funct., 21 (2010), 327-336. doi: 10.1080/10652460903259915
    [66] F. Qi, A note on Schur-convexity of extended mean values, Rocky Mountain J. Math., 35 (2005), 1787-1793. doi: 10.1216/rmjm/1181069663
    [67] F. Qi, Integral representations and properties of Stirling numbers of the first kind, J. Number Theory, 133 (2013), 2307-2319. doi: 10.1016/j.jnt.2012.12.015
    [68] F. Qi, Logarithmic convexity of extended mean values, Proc. Amer. Math. Soc., 130 (2002), 1787-1796. doi: 10.1090/S0002-9939-01-06275-X
    [69] F. Qi, C. Berg, Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function, Mediterr. J. Math., 10 (2013), 1685-1696. doi: 10.1007/s00009-013-0272-2
    [70] F. Qi, P. Cerone, S. S. Dragomir, et al. Alternative proofs for monotonic and logarithmically convex properties of one-parameter mean values, Appl. Math. Comput. 208 (2009), 129-133.
    [71] F. Qi, J. X. Cheng, Some new Steffensen pairs, Anal. Math., 29 (2003), 219-226. doi: 10.1023/A:1025467221664
    [72] F. Qi, B. N. Guo, On Steffensen pairs, J. Math. Anal. Appl., 271 (2002), 534-541. doi: 10.1016/S0022-247X(02)00120-8
    [73] F. Qi, B. N. Guo, Some properties of extended remainder of Binet's first formula for logarithm of gamma function, Math. Slovaca, 60 (2010), 461-470. doi: 10.2478/s12175-010-0025-7
    [74] F. Qi, S. L. Xu, The function (bx - ax)/x: inequalities and properties, Proc. Amer. Math. Soc., 126 (1998), 3355-3359. doi: 10.1090/S0002-9939-98-04442-6
    [75] S. Q. Zhang, B. N. Guo, F. Qi, A concise proof for properties of three functions involving the exponential function, Appl. Math. E-Notes, 9 (2009), 177-183.
    [76] F. Qi, Q. M. Luo, B. N. Guo, The function (bx - ax)/x: Ratio's properties, In: Analytic Number Theory, Approximation Theory, and Special Functions, G. V. Milovanović, M. Th. Rassias (Eds), Springer, 2014, 485-494.
    [77] H. Alzer, Complete monotonicity of a function related to the binomial probability, J. Math. Anal. Appl., 459 (2018), 10-15. doi: 10.1016/j.jmaa.2017.10.077
    [78] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics-A Foundation for Computer Science, 2Eds., Addison-Wesley Publishing Company, Reading, MA, 1994.
    [79] B. N. Guo, F. Qi, Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math., 272 (2014), 251-257. doi: 10.1016/j.cam.2014.05.018
    [80] B. N. Guo, F. Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, J. Comput. Appl. Math., 255 (2014), 568-579. doi: 10.1016/j.cam.2013.06.020
    [81] F. Ouimet, Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex, J. Math. Anal. Appl., 466 (2018), 1609-1617. doi: 10.1016/j.jmaa.2018.06.049
    [82] F. Qi, A logarithmically completely monotonic function involving the q-gamma function, HAL preprint, 2018, Available from: https://hal.archives-ouvertes.fr/hal-01803352v1.
    [83] F. Qi, Complete monotonicity for a new ratio of finite many gamma functions, HAL preprint, 2020, Available from: https://hal.archives-ouvertes.fr/hal-02511909v1.
    [84] F. Qi, B. N. Guo, From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions, arXiv preprint, 2020, Available from: https://arxiv.org/abs/2001.02175v1.
    [85] F. Qi, W. H. Li, S. B. Yu, et al. A ratio of many gamma functions and its properties with applications, arXiv preprint, 2019, Available from: https://arXiv.org/abs/1911.05883v1.
    [86] F. Qi, D. Lim, Monotonicity properties for a ratio of finite many gamma functions, HAL preprint, 2020, Available from: https://hal.archives-ouvertes.fr/hal-02511883v1.
    [87] F. Qi, D. W. Niu, D. Lim, et al. Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions, HAL preprint, 2018, Available from: https://hal.archives-ouvertes.fr/hal-01769288v1.
    [88] C. F. Wei, B. N. Guo, Complete monotonicity of functions connected with the exponential function and derivatives, Abstr. Appl. Anal., 2014 (2014), Article ID 851213, 5.
    [89] A. M. Xu, Z. D. Cen, Some identities involving exponential functions and Stirling numbers and applications, J. Comput. Appl. Math., 260 (2014), 201-207. doi: 10.1016/j.cam.2013.09.077
    [90] B. N. Guo, F. Qi, An alternative proof of Elezović-Giordano-Pečarić's theorem, Math. Inequal. Appl., 14 (2011), 73-78.
    [91] F. Qi, B. N. Guo, C. P. Chen, The best bounds in Gautschi-Kershaw inequalities, Math. Inequal. Appl., 9 (2006), 427-436.
    [92] J. L. Zhao, Q. M. Luo, B. N. Guo, et al. Logarithmic convexity of Gini means, J. Math. Inequal., 6 (2012), 509-516. doi: 10.7153/jmi-06-48
    [93] F. Qi, Completely monotonic degree of a function involving the tri- and tetra-gamma functions, arXiv preprint, 2013, Available from: http://arxiv.org/abs/1301.0154v1.
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