Completely monotonic degree of a function involving trigamma and tetragamma functions

Feng Qi; Feng Qi

doi:10.3934/math.2020219

AIMS Mathematics

2020, Volume 5, Issue 4: 3391-3407. doi: 10.3934/math.2020219

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Research article

Completely monotonic degree of a function involving trigamma and tetragamma functions

Feng Qi ^{1,2,3
,
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1.
Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, Henan, China
2.
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China
3.
School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China

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 $\Psi(x)$.
- completely monotonic degree,
- completely monotonic function,
- trigamma function,
- tetragamma function,
- strongly completely monotonic function,
- logarithmically concave function,
- convolution,
- open problem
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

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Abstract

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 $\Psi(x)$.

References

[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 (b^x - a^x)/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 (b^x - a^x)/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 (b^x - a^x)/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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