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DOI: 10.32604/cmes.2022.016927
ARTICLE
Complete Monotonicity of Functions Related to Trigamma and Tetragamma Functions
1Faculty of Science, Mathematics Department, Mansoura University, Mansoura, 35516, Egypt
2Faculty of Science, Mathematics Department, Jeddah University, Jeddah, 21589, Saudi Arabia
3Faculty of Science, Mathematics Department, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
*Corresponding Author: Mansour Mahmoud. Email: mansour@mans.edu.eg
Received: 11 April 2021; Accepted: 20 October 2021
Abstract: In this paper, we study the completely monotonic property of two functions involving the function
Keywords: Trigamma function; tetragamma function; completely monotonic function; completely monotonic degree; inequality
The Euler’s gamma function is defined [1] by the improper integral
and the psi or digamma function is defined by the logarithmic derivative of gamma function, that is
The two derivatives
Kirchhoff was the first to apply the polygamma functions
A function f(x) is said to be completely monotonic [6] (Chapter XIII) on an interval J if the derivatives f(m)(x) exist on J for all
According to the Bernstein–Widder theorem [7] the function f(x) is completely monotonic on
where the integral converges for
In 2004, Alzer [8] presented the inequality
where
and
In 2013, Guo et al. [9] proved the complete monotonicity on
In [10], Zhao et al. proved the complete monotonicity on
and
They also presented the double inequality
The lower bound of inequality (2) and the bound of inequality (1) are not included each other.
In 2015, Qi [11] proved the complete monotonicity on
If and only if
If and only if
The survey [12] presented several results about the function
Let the function f (x) be completely monotonic for
Qi [27] proved that
and Guo et al. [26] proved that
where C(x) is a polynomial of positive coefficients of degree twenty one. For more results related to properties of completely monotonic degree, we refer to [12,16,28–32] and the references therein.
Our first aim will be to establish the double inequality
which improves the upper bound of inequality (3) for x > 0 and the lower bound of inequality (3) for
and
The second aim of this paper is to compute the completely monotonic degree of a function involving
For proving our main results, we need the following lemmas.
Lemma 2.1 The function
is completely monotonic on
where
Proof. Using the integral representations formulas [33]
and
we have
where
with
Using the inequalities 3n > n2 for
Also, by using the inequalities
and hence an > 0 for
where the mth Bernoulli number Bm is defined by [34]
We obtain
Lemma 2.2 The function
is completely monotonic on
Proof. Using the formulas (6) and (7), we obtain
where
with
Now
By using that
For
Hence bn < 0 for
Now we begin to prove our main results.
Theorem 3.1 The functions
and
are completely monotonic on
Proof. Using recursion formula [33]
we get
The two functions
We get
Now, using the recursion formula (16), we get
The two functions
The proof of Theorem 1 is complete.
Remark 1. The upper bound of inequality (15) is better than the upper bound of inequality (3) for x > 0. Also, the lower bound of inequality (15) is better than the lower bound of inequality (3) for
Theorem 3.2. The completely monotonic degree of L(x) on
Proof. Using the integral representation (8), we get
where
with
Now using the inequalities
we obtain that kn > 0 for
If we suppose that
where
with
Using the relation [35]
We get
Also,
Then
and hence we get
Combining (18) and (19) completes the proof.
The main conclusions of this paper are stated in Theorems 3.1 and 3.2. Concretely speaking, the authors proved the completely monotonic property of two functions involving the sum of the Trigamma square and Tetragamma functions, derived a new double inequality for this sum and deduced the completely monotonic degree of a function involving the Trigamma function.
Funding Statement: The authors received no specific funding for this study.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
1. Andrews, G. E., Askey, R. A., Roy, R. (1999). Special functions, encyclopedia of mathematics and its applications, vol. 71. Cambridge: Cambridge University Press. [Google Scholar]
2. Coffey, M. W. (2006). One integral in three ways: Moments of a quantum distribution. Journal of Physics A: Mathematical and General, 39, 1425–1431. DOI 10.1088/0305-4470/39/6/015. [Google Scholar] [CrossRef]
3. Miller, A. R. (2006). Summations for certain series containing the digamma function. Journal of Physics A: Mathematical and General, 39, 3011–3020. DOI 10.1088/0305-4470/39/12/010. [Google Scholar] [CrossRef]
4. Wilkins, B. D., Hromadka, T. V. (2021). Using the digamma function for basis functions in mesh-free computational methods. Engineering Analysis with Boundary Elements, 131, 218–227. DOI 10.1016/j.enganabound.2021.06.004. [Google Scholar] [CrossRef]
5. Qiu, S. L., Vuorinen, M. (2005). Some properties of the gamma and psi functions, with applications. Mathematics of Computation, 74(250), 723–742. DOI 10.1090/S0025-5718-04-01675-8. [Google Scholar] [CrossRef]
6. Mitrinović, D. S., Pečarić, J. E., Fink, A. M. (1993). Classical and new Inequalities in Analysis, Dordrecht-Boston-London: Kluwer Academic Publishers. [Google Scholar]
7. Widder, D. V. (1946). The laplace transform. Princeton: Princeton University Press. [Google Scholar]
8. Alzer, H. (2004). Sharp inequalities for the digamma and polygamma functions. Forum Mathematicum, 16(2), 181–221. DOI 10.1515/form.2004.009. [Google Scholar] [CrossRef]
9. Guo, B. N., Zhao, J. L., Qi, F. (2013). A completely monotonic function involving the tri- and tetra-gamma functions. Mathematica Slovaca, 63(3), 469–478. DOI 10.2478/s12175-013-0109-2. [Google Scholar] [CrossRef]
10. Zhao, J. L., Guo, B. N., Qi, F. (2012). Complete monotonicity of two functions involving the tri- and tetra-gamma functions. Periodica Mathematica Hungarica, 65(1), 147–155. DOI 10.1007/s10998-012-9562-x. [Google Scholar] [CrossRef]
11. Qi, F. (2015). Complete monotonicity of a function involving the tri- and tetra-gamma functions. Proceedings of the Jangjeon Mathematical Society, 18(2), 253–264. DOI 10.17777/pjms.2015.18.2.25. [Google Scholar] [CrossRef]
12. Qi, F., Agarwal, R. P. (2019). On complete monotonicity for several classes of functions related to ratios of gamma functions. Journal of Inequalities and Applications, 2019, 36, 1–42. DOI 10.1186/s13660-019-1976-z. [Google Scholar] [CrossRef]
13. Batir, N. (2007). On some properties of digamma and polygamma functions. Journal of Mathematical Analysis and Applications, 328(1), 452–465. DOI 10.1016/j.jmaa.2006.05.065. [Google Scholar] [CrossRef]
14. Guo, B. N., Qi, F., Srivastava, H. M. (2010). Some uniqueness results for the non-trivially complete monotonicity of a class of functions involving the polygamma and related functions. Integral Transforms and Special Functions, 21(11), 849–858. DOI 10.1080/10652461003748112. [Google Scholar] [CrossRef]
15. Guo, B. N., Qi, F. (2011). A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications. Journal of the Korean Mathematical Society, 48(3), 655–667. DOI 10.4134/JKMS.2011.48.3.655. [Google Scholar] [CrossRef]
16. Koumandos, S. (2008). Monotonicity of some functions involving the gamma and psi functions. Mathematics of Computation, 77(264), 2261–2275. DOI 10.1090/S0025-5718-08-02140-6. [Google Scholar] [CrossRef]
17. Qi, F. (2014). A completely monotonic function related to the q-trigamma function. University Politehnica of Bucharest Scientific Bulletin-Series A–Applied Mathematics and Physics, 76(1), 107–114. [Google Scholar]
18. Qi, F. (2015). Complete monotonicity of functions involving the q-trigamma and q-tetragamma functions. Revista de la Real Academia de Ciencias Exactas, F¡́x0131/¿sicas y Naturales, Serie A Matemáticas, Journal of the Spanish Royal Academy of Sciences, Series A Mathematics,, 109(2), 419–429. DOI 10.1007/s13398-014-0193-3. [Google Scholar] [CrossRef]
19. Qi, F., Guo, B. N. (2016). Complete monotonicity of divided differences of the di- and tri-gamma functions with applications. Georgian Mathematical Journal, 23(2), 279–291. DOI 10.1515/gmj-2016-0004. [Google Scholar] [CrossRef]
20. Qi, F., Liu, F. F., Shi, X. T. (2016). Comments on two completely monotonic functions involving the q-trigamma function. Journal of Inequalities and Special Functions, 7(4), 211–217. [Google Scholar]
21. Qi, F. (2021). Necessary and sufficient conditions for a difference constituted by four derivatives of a function involving trigamma function to be completely monotonic. Mathematical Inequalities and Applications, 24(3), 845–855. DOI 10.7153/mia-2021-24-58. [Google Scholar] [CrossRef]
22. Qi, F. (2021). Necessary and sufficient conditions for a ratio involving trigamma and tetragamma functions to be monotonic. Turkish Journal of Inequalities, 5(1), 50–59. [Google Scholar]
23. Qi, F. (2021). Necessary and sufficient conditions for complete monotonicity and monotonicity of two functions defined by two derivatives of a function involving trigamma function. Applicable Analysis and Discrete Mathematics, 15(2), 378–392. DOI 10.2298/AADM191111014Q. [Google Scholar] [CrossRef]
24. Qi, F. (2021). Necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic. TWMS Journal of Pure and Applied Mathematics, 12(2), (in Press). [Google Scholar]
25. Zhao, J. L. (2015). A completely monotonic function relating to the q-trigamma function. Journal of Mathematical Inequalities, 9(1), 53–60. DOI 10.7153/jmi-09-05. [Google Scholar] [CrossRef]
26. Guo, B. N., Qi, F. (2012). A completely monotonic function involving the tri-gamma function and with degree one. Applied Mathematics and Computation, 218(19), 9890–9897. DOI 10.1016/j.amc.2012.03.075. [Google Scholar] [CrossRef]
27. Qi, F. (2020). Completely monotonic degree of a function involving trigamma and tetragamma functions. AIMS Mathematics, 5(4), 3391–3407. DOI 10.3934/math.2020219. [Google Scholar] [CrossRef]
28. Guo, B. N., Qi, F. (2015). On the degree of the weighted geometric mean as a complete Bernstein function. Afrika Matematika, 26(7), 1253–1262. DOI 10.1007/s13370-014-0279-2. [Google Scholar] [CrossRef]
29. Koumandos, S., Lamprecht, M. (2013). Complete monotonicity and related properties of some special functions. Mathematics of Computation, 82(282), 1097–1120. DOI 10.1090/S0025-5718-2012-02629-9. [Google Scholar] [CrossRef]
30. Qi, F., Wang, S. H. (2014). Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions. Global Journal of Mathematical Analysis, 2(3), 91–97. DOI 10.14419/gjma.v2i3.2919. [Google Scholar] [CrossRef]
31. Qi, F. (2015). Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions. Mathematical Inequalities and Applications, 18(2), 493–518. DOI 10.7153/mia-18-37. [Google Scholar] [CrossRef]
32. Qi, F., Liu, A. Q. (2019). Completely monotonic degrees for a difference between the logarithmic and psi functions. Journal of Computational and Applied Mathematics, 361, 366–371. DOI 10.1016/j.cam.2019.05.001. [Google Scholar] [CrossRef]
33. Abramowitz, M., Stegun, I. A. (Eds). (1972). Handbook of mathematical functions with formulas, graphs, and mathematical tables (9th ed), vol. 55. Washington DC: National Bureau of Standards, Applied Mathematics Series. [Google Scholar]
34. Shuang, Y., Guo, B. N., Qi, F. (2021). Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios. Revista de la Real Academia de Ciencias Exactas, F¡́x0131/¿sicas y Naturales, Serie A. Matemáticas, 115(3), 1–12. DOI 10.1007/s13398-021-01071-x. [Google Scholar] [CrossRef]
35. Alzer, H. (1997). On some inequalities for the gamma and psi functions. Mathematics of Computation, 66(217), 373–389. DOI 10.1090/S0025-5718-97-00807-7. [Google Scholar] [CrossRef]
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