AIMS Mathematics, 2020, 5(4): 3391-3407. doi: 10.3934/math.2020219.

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Completely monotonic degree of a function involving trigamma and tetragamma functions

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

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

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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 21. F. Qi, D. Lim, Integral representations of bivariate complex geometric mean and their applications, J. Comput. Appl. Math., 330 (2018), 41-58.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 37. N. Batir, On some properties of digamma and polygamma functions, J. Math. Anal. Appl., 328 (2007), 452-465.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 60. B. N. Guo, F. Qi, A simple proof of logarithmic convexity of extended mean values, Numer. Algorithms, 52 (2009), 89-92.
• 61. B. N. Guo, F. Qi, Generalization of Bernoulli polynomials, Int. J. Math. Ed. Sci. Tech., 33 (2002), 428-431.
• 62. B. N. Guo, F. Qi, Properties and applications of a function involving exponential functions, Commun. Pure Appl. Anal., 8 (2009), 1231-1249.
• 63. B. N. Guo, F. Qi, The function (bx - ax)/x: Logarithmic convexity and applications to extended mean values, Filomat, 25 (2011), 63-73.
• 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.
• 66. F. Qi, A note on Schur-convexity of extended mean values, Rocky Mountain J. Math., 35 (2005), 1787-1793.
• 67. F. Qi, Integral representations and properties of Stirling numbers of the first kind, J. Number Theory, 133 (2013), 2307-2319.
• 68. F. Qi, Logarithmic convexity of extended mean values, Proc. Amer. Math. Soc., 130 (2002), 1787-1796.
• 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.
• 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.
• 72. F. Qi, B. N. Guo, On Steffensen pairs, J. Math. Anal. Appl., 271 (2002), 534-541.
• 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.
• 74. F. Qi, S. L. Xu, The function (bx - ax)/x: inequalities and properties, Proc. Amer. Math. Soc., 126 (1998), 3355-3359.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.