Inequalities, asymptotic expansions and complete monotonicity properties related to the gamma and digamma functions

Shun-Wei Xu; Chao-Ping Chen; H. M. Srivastava; Cristinel Mortici

doi:10.56754/0719-0646.2803.503

Authors

Shun-Wei Xu School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo Henan 454003, China https://orcid.org/0009-0003-3669-0461
Chao-Ping Chen School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo Henan 454003, China https://orcid.org/0000-0002-8263-9216
H. M. Srivastava Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada; Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China https://orcid.org/0000-0002-9277-8092
Cristinel Mortici Valahia University of Târgovişte, Aleea Sinaia 13, RO-130004 Târgovişte, Romania; Academy of Romanian Scientists, Splaiul Independenţei 54, RO-050085 Bucharest, Romania https://orcid.org/0000-0001-5576-1921

MSC:

33B15, 41A60, 26D07, 26A48

DOI:

https://doi.org/10.56754/0719-0646.2803.503

Keywords:

Gamma function, Digamma (or psi) function, Asymptotic expansions, Inequalities, Completely monotonic functions

Abstract

The quotient of two gamma functions has the following known asymptotic expansion:

\(\displaystyle \frac{\Gamma(x+a)}{\Gamma(x+b)}\sim x^{a-b}\sum_{k=0}^{\infty}\binom{a-b}{k}B_{k}^{(a-b+1)}(a)\,x^{-k}\quad(x\to\infty),\)

where \(B_{k}^{(\nu)}(x)\) denotes the generalized Bernoulli polynomials. In this article, without using these generalized Bernoulli polynomials \(B_{k}^{(\nu)}(x)\), we present two asymptotic expansions and a recurrence relation for determining the coefficients of each asymptotic expansion related to the quotient \(\frac{\Gamma(x+a)}{\Gamma(x+b)}\) of two gamma functions. Let

\(\displaystyle \Omega_{n}:=\frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2}+1\right)},\quad n\in\mathbb{N}\)

be the volume of the unit ball in \(\mathbb{R}^{n}\). We establish a sharp two-sided inequality for the quotient \(\frac{\Omega_{n-1}}{\Omega_{n}}\). We also present a complete monotonicity property for the Nielsen's \(\beta\)-function, which extends a known result due to Matejíčka.

References

[1] H. Alzer, “Inequalities for the volume of the unit ball in R^n,” J. Math. Anal. Appl., vol. 252, no. 1, pp. 353–363, 2000, doi: 10.1006/jmaa.2000.7065.

[2] H. Alzer, “Inequalities for the volume of the unit ball in R^n. II,” Mediterr. J. Math., vol. 5, no. 4, pp. 395–413, 2008, doi: 10.1007/s00009-008-0158-x.

[3] H. Alzer and G. Jameson, “A harmonic mean inequality for the digamma function and related results,” Rend. Semin. Mat. Univ. Padova, vol. 137, pp. 203–209, 2017, doi: 10.4171/RSMUP/137-10.

[4] G. D. Anderson and S.-L. Qiu, “A monotoneity property of the gamma function,” Proc. Am. Math. Soc., vol. 125, no. 11, pp. 3355–3362, 1997, doi: 10.1090/S0002-9939-97-04152-X.

[5] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, “Special functions of quasiconformal theory,” Expo. Math., vol. 7, no. 2, pp. 97–136, 1989.

[6] T. Ban and C.-P. Chen, “New inequalities for the volume of the unit ball in R^n,” J. Math. Inequal., vol. 11, no. 2, pp. 527–542, 2017, doi: 10.7153/jmi-11-43.

[7] C. Berg, S. Koumandos, and H. L. Pedersen, “Nielsen’s beta function and some infinitely divisible distributions,” Math. Nachr., vol. 294, no. 3, pp. 426–449, 2021, doi: 10.1002/mana.201900217.

[8] C. Berg and H. L. Pedersen, “A one-parameter family of Pick functions defined by the Gamma function and related to the volume of the unit ball in n-space,” Proc. Am. Math. Soc., vol. 139, no. 6, pp. 2121–2132, 2011, doi: 10.1090/S0002-9939-2010-10636-6.

[9] K. H. Borgwardt, The simplex method. A probabilistic analysis, ser. Algorithms Comb. Berlin: Springer, 1987, vol. 1.

[10] C. A. Charalambides, Enumerative combinatorics, ser. CRC Press Ser. Discrete Math. Appl. Boca Raton, FL: Chapman & Hall/CRC, 2002.

[11] C.-P. Chen and L. Lin, “Inequalities for the volume of the unit ball in R^n,” Mediterr. J. Math., vol. 11, no. 2, pp. 299–314, 2014, doi: 10.1007/s00009-013-0340-7.

[12] C.-P. Chen and R. B. Paris, “Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function,” Appl. Math. Comput., vol. 250, pp. 514–529, 2015, doi: 10.1016/j.amc.2014.11.010.

[13] C.-P. Chen and R. B. Paris, “Inequalities and asymptotic expansions related to the volume of the unit ball in R^n,” Result. Math., vol. 74, no. 1, 2019, Art. ID 44, doi: 10.1007/s00025-019-0967-1.

[14] C.-P. Chen and F. Qi, “The best bounds in Wallis’ inequality,” Proc. Am. Math. Soc., vol. 133, no. 2, pp. 397–401, 2005, doi: 10.1090/S0002-9939-04-07499-4.

[15] C.-P. Chen, H. M. Srivastava, L. Li, and S. Manyama, “Inequalities and monotonicity properties for the psi (or digamma) function and estimates for the Euler-Mascheroni constant,” Integral Transforms Spec. Funct., vol. 22, no. 9, pp. 681–693, 2011, doi: 10.1080/10652469.2010.538525.

[16] L. Comtet, Advanced Combinatorics. Dordrecht: D. Reidel Publishing Company, 1974.

[17] D. Cvijović, “New identities for the partial Bell polynomials,” Appl. Math. Lett., vol. 24, no. 9, pp. 1544–1547, 2011, doi: 10.1016/j.aml.2011.03.043.

[18] M. J. Dubourdieu, “Sur un théorème de M. S. Bernstein rélatif à la transformation de Laplace-Stieltjes,” Compos. Math., vol. 7, pp. 96–111, 1939.

[19] W. Gautschi, “A harmonic mean inequality for the gamma function,” SIAM J. Math. Anal., vol. 5, pp. 278–281, 1974, doi: 10.1137/0505030.

[20] B.-N. Guo and F. Qi, “Monotonicity and logarithmic convexity relating to the volume of the unit ball,” Optim. Lett., vol. 7, no. 6, pp. 1139–1153, 2013, doi: 10.1007/s11590-012-0488-2.

[21] D. A. Klain and G.-C. Rota, “A continuous analogue of Sperner’s theorem,” Commun. Pure Appl. Math., vol. 50, no. 3, pp. 205–223, 1997, doi: 10.1002/(SICI)1097-0312(199703)50:3<205::AID-CPA1>3.0.CO;2-F.

[22] S.-D. Lin, T.-M. Hsieh, and H. M. Srivastava, “Some families of multiple infinite sums associated with the digamma and related functions,” J. Fractional Calc., vol. 24, pp. 77–85, 2003.

[23] D. Lu and P. Zhang, “A new general asymptotic formula and inequalities involving the volume of the unit ball,” J. Number Theory, vol. 170, pp. 302–314, 2017, doi: 10.1016/j.jnt.2016.06.010.

[24] M. Masjed-Jamei, Z. Moalemi, W. Koepf, and H. M. Srivastava, “An extension of the Taylor series expansion by using the Bell polynomials,” Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM, vol. 113, no. 2, pp. 1445–1461, 2019, doi: 10.1007/s13398-018-0558-0.

[25] L. Matejíčka, “Proof of a conjecture on Nielsen’s β-function,” Probl. Anal. Issues Anal., vol. 8, no. 3, pp. 105–111, 2019, doi: 10.15393/j3.art.2019.6810.

[26] M. Merkle, “Gurland’s ratio for the gamma function,” Comput. Math. Appl., vol. 49, no. 2-3, pp. 389–406, 2005, doi: 10.1016/j.camwa.2004.01.016.

[27] C. Mortici, “Monotonicity properties of the volume of the unit ball in R^n,” Optim. Lett., vol. 4, no. 3, pp. 457–464, 2010, doi: 10.1007/s11590-009-0173-2.

[28] C. Mortici, “Series associated to some expressions involving the volume of the unit ball and applications,” Appl. Math. Comput., vol. 294, pp. 121–138, 2017, doi: 10.1016/j.amc.2016.08.027.

[29] K. Nantomah, M. Iddrisu, and C. Okpoti, “On a q-analogue of the Nielsen’s β-function,” Int. J. Math. Appl., vol. 6, no. 2 - A, p. 163–171, 2018.

[30] K. Nantomah, “Monotonicity and convexity properties and some inequalities involving a generalized form of the Wallis’ cosine formula,” Asian Res. J. Math., vol. 6, no. 3, p. 1–10, 2017, doi: 10.9734/ARJOM/2017/36699.

[31] K. Nantomah, “Monotonicity and convexity properties of the Nielsen’s β-function,” Probl. Anal. Issues Anal., vol. 6, no. 2, pp. 81–93, 2017, doi: 10.15393/j3.art.2017.3950.

[32] K. Nantomah, “A generalization of the Nielsen’s β-function,” Int. J. Math. Appl, vol. 11, pp. 16–26, 2018, doi: 10.12816/0049058.

[33] K. Nantomah, “On some properties and inequalities for the Nielsen’s β-function,” Sci., Ser. A, Math. Sci. (N.S.), vol. 28, pp. 43–54, 2018.

[34] K. Nantomah, “Certain properties of the Nielsen’s β-function,” Bull. Int. Math. Virtual Inst., vol. 9, no. 2, pp. 263–269, 2019.

[35] K. Nantomah, “New inequalities for Nielsen’s beta function,” Commun. Math. Appl., vol. 10, pp. 773–781, 2019, doi: 10.26713/cma.v10i4.1233.

[36] K. Nantomah, “An alternative proof of a harmonic mean inequality for Nielsen’s beta function,” Contrib. Math., vol. 4, pp. 12–13, 2021, doi: 10.47443/cm.2021.0028.

[37] N. Nielsen, Handbuch der Theorie der Gammafunktion. Leipzig: Teubner, 1906.

[38] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Eds., NIST handbook of mathematical functions. Cambridge: Cambridge University Press, 2010.

[39] A. Prabhu and H. M. Srivastava, “Some limit formulas for the gamma and psi (or digamma) functions at their singularities,” Integral Transforms Spec. Funct., vol. 22, no. 8, pp. 587–592, 2011, doi: 10.1080/10652469.2010.535970.

[40] J. Riordan, “Combinatorial identities,” New York-London-Sydney: John Wiley and Sons, Inc., 1968.

[41] S. Rota Bulò, E. R. Hancock, F. Aziz, and M. Pelillo, “Efficient computation of Ihara coefficients using the Bell polynomial recursion,” Linear Algebra Appl., vol. 436, no. 5, pp. 1436–1441, 2012, doi: 10.1016/j.laa.2011.08.017.

[42] H. van Haeringen, “Completely monotonic and related functions,” J. Math. Anal. Appl., vol. 204, no. 2, pp. 389–408, 1996, doi: 10.1006/jmaa.1996.0443.

[43] L. Yin and L.-G. Huang, “Some inequalities for the volume of the unit ball,” J. Class. Anal., vol. 6, no. 1, pp. 39–46, 2015, doi: 10.7153/jca-06-03.

[44] H. Zhang, “New bounds and asymptotic expansions for the volume of the unit ball in R^n based on Padé approximation,” Result. Math., vol. 77, no. 3, 2022, Art. ID 116, doi: 10.1007/s00025-022-01652-1.

[45] J. Zhang, L. Yin, and W. Cui, “Some properties of generalized Nielsen’s β-function with double parameters,” Turk. J. Anal. Number Theory, vol. 2, pp. 37–43, 2018, doi: 10.12691/tjant-7-1-4.

[46] J. Zhang, L. Yin, and W. Cui, “Monotonic properties of generalized Nielsen’s β-function,” Turk. J. Anal. Number Theory, vol. 7, pp. 18–22, 2019, doi: 10.12691/tjant-7-1-4.