A LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTION INVOLVING THE RATIO OF GAMMA FUNCTIONS

Feng Qi; Wen-Hui Li

doi:10.11948/2015049

2015 Volume 5 Issue 4

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2015 > 5(4): 626-634

Next Article Previous Article

Feng Qi, Wen-Hui Li. A LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTION INVOLVING THE RATIO OF GAMMA FUNCTIONS[J]. Journal of Applied Analysis & Computation, 2015, 5(4): 626-634. doi: 10.11948/2015049

Citation:

Feng Qi, Wen-Hui Li. A LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTION INVOLVING THE RATIO OF GAMMA FUNCTIONS[J]. Journal of Applied Analysis & Computation, 2015, 5(4): 626-634. doi: 10.11948/2015049

A LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTION INVOLVING THE RATIO OF GAMMA FUNCTIONS

Feng Qi^1,2,3,
Wen-Hui Li³

1 Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China;
2 College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China;
3 Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, 300387, China

Fund Project:

Abstract

Abstract

In the paper, the authors concisely survey and review some functions involving the gamma function and its various ratios, simply state their logarithmically complete monotonicity and related results, and find necessary and sufficient conditions for a new function involving the ratio of two gamma functions and originating from the coding gain to be logarithmically completely monotonic.
- Logarithmically completely monotonic function /
- ratio /
- gamma function /
- coding gain /
- necessary and sufficient condition
MSC: 33B15

FullText(HTML)

References

[1]	M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Washington, 1972. Google Scholar
[2]	R. D. Atanassov and U. V. Tsoukrovski, Some properties of a class of logarithmically completely monotonic functions, C. R. Acad. Bulgare Sci., 41(2)(1988), 21-23. Google Scholar
[3]	C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math., 1(4)(2004), 433-439. Google Scholar
[4]	T. Burić and N. Elezović, New asymptotic expansions of the quotient of gamma functions, Integral Transforms Spec. Funct., 23(5)(2012), 355-368. Google Scholar
[5]	M. W. Coffey, Fractional part integral representation for derivatives of a function related to ln Γ(x +1), Publ. Math. Debrecen, 80(3-4)(2012), 347-358. Google Scholar
[6]	B.-N. Guo and 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(3)(2011), 655-667. Google Scholar
[7]	B.-N. Guo and F. Qi, A class of completely monotonic functions involving the gamma and polygamma functions, Cogent Math., 1(2014), 1:982896, 8 pages. Google Scholar
[8]	B.-N. Guo and F. Qi, A completely monotonic function involving the tri-gamma function and with degree one, Appl. Math. Comput., 218(19)(2012), 9890-9897. Google Scholar
[9]	B.-N. Guo and F. Qi, A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 72(2)(2010), 21-30. Google Scholar
[10]	B.-N. Guo and F. Qi, An extension of an inequality for ratios of gamma functions, J. Approx. Theory, 163(9)(2011), 1208-1216. Google Scholar
[11]	B.-N. Guo and F. Qi, Inequalities and monotonicity for the ratio of gamma functions, Taiwanese J. Math., 7(2)(2003), 239-247. Google Scholar
[12]	B.-N. Guo and F. Qi, Monotonicity and logarithmic convexity relating to the volume of the unit ball, Optim. Lett., 7(6)(2013), 1139-1153. Google Scholar
[13]	B.-N. Guo and F. Qi, Monotonicity of functions connected with the gamma function and the volume of the unit ball, Integral Transforms Spec. Funct., 23(9)(2012), 701-708. Google Scholar
[14]	B.-N. Guo and F. Qi, Refinements of lower bounds for polygamma functions, Proc. Amer. Math. Soc., 141(3)(2013), 1007-1015. Google Scholar
[15]	B.-N. Guo and F. Qi, Sharp inequalities for the psi function and harmonic numbers, Analysis (Berlin), 34(2)(2014), 201-208. Google Scholar
[16]	B.-N. Guo, J.-L. Zhao, and F. Qi,A completely monotonic function involving the tri-and tetra-gamma functions, Math. Slovaca, 63(3)(2013), 469-478. Google Scholar
[17]	S. Koumandos, On Ruijsenaars' asymptotic expansion of the logarithm of the double gamma function, J. Math. Anal. Appl., 341(2008), 1125-1132. Google Scholar
[18]	S. Koumandos and M. Lamprecht, Complete monotonicity and related properties of some special functions, Math. Comp., 82(282)(2013), 1097-1120. Google Scholar
[19]	S. Koumandos and 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(1)(2009), 33-40. Google Scholar
[20]	V. Krasniqi and F. Qi, Complete monotonicity of a function involving the p-psi function and alternative proofs, Glob. J. Math. Anal., 2(3)(2014), 204-208. Google Scholar
[21]	J. Lee and C. Tepedelenlioǧlu, Space-time coding over fading channels with stable noise, IEEE Transactions on Vehicular Technology, 60(7)(2011), 3169-3177. Google Scholar
[22]	W.-H. Li, F. Qi, and B.-N. Guo, On proofs for monotonicity of a function involving the psi and exponential functions, Analysis (Munich), 33(1)(2013), 45-50. Google Scholar
[23]	F. Qi, A completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums, ANZIAM J., 48(4)(2007), 523-532. Google Scholar
[24]	F. Qi, A completely monotonic function related to the q-trigamma function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 76(1)(2014), 107-114. Google Scholar
[25]	F. Qi, An integral representation, complete monotonicity, and inequalities of Cauchy numbers of the second kind, J. Number Theory, 144(2014), 244-255. Google Scholar
[26]	F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl., 2010(2010), Article ID 493058, 84 pages. Google Scholar
[27]	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(5)(2014), 152-164. Google Scholar
[28]	F. Qi, Complete monotonicity of a function involving the tri-and tetra-gamma functions, Proc. Jangjeon Math. Soc., 18(2)(2015), 253-264. Google Scholar
[29]	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. Google Scholar
[30]	F. Qi, Integral representations and properties of Stirling numbers of the first kind, J. Number Theory, 133(7)(2013), 2307-2319. Google Scholar
[31]	F. Qi, Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions, Math. Inequal. Appl., 18(2)(2015), 493-518. Google Scholar
[32]	F. Qi, Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct., 18(7)(2007), 503-509. Google Scholar
[33]	F. Qi and 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(4)(2013), 1685-1696. Google Scholar
[34]	F. Qi, P. Cerone, and S. S. Dragomir, Complete monotonicity of a function involving the divided difference of psi functions, Bull. Aust. Math. Soc., 88(2)(2013), 309-319. Google Scholar
[35]	F. Qi and C.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl., 296(2004), 603-607. Google Scholar
[36]	F. Qi and B.-N. Guo, A logarithmically completely monotonic function involving the gamma function, Taiwanese J. Math., 14(4)(2010), 1623-1628. Google Scholar
[37]	F. Qi and B.-N. Guo,Complete monotonicity of divided differences of the diand tri-gamma functions and applications, Georgian Math. J., 22(2015), in press. Google Scholar
[38]	F. Qi and B.-N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll., 7(1)(2004), Art. 8, 63-72. Google Scholar
[39]	F. Qi and B.-N. Guo, Completely monotonic functions involving divided differences of the di-and tri-gamma functions and some applications, Commun. Pure Appl. Anal., 8(6)(2009), 1975-1989. Google Scholar
[40]	F. Qi and B.-N. Guo, Monotonicity and convexity of ratio between gamma functions to different powers, J. Indones. Math. Soc. (MIHMI), 11(1)(2005), 39-49. Google Scholar
[41]	F. Qi and B.-N. Guo, Some logarithmically completely monotonic functions related to the gamma function, J. Korean Math. Soc., 47(6)(2010), 1283-1297. Google Scholar
[42]	F. Qi, B.-N. Guo, and C.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, RGMIA Res. Rep. Coll., 7(1)(2004), Art. 5, 31-36. Google Scholar
[43]	F. Qi, B.-N. Guo, and C.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, J. Aust. Math. Soc., 80(2006), 81-88. Google Scholar
[44]	F. Qi, B.-N. Guo, S. Guo, and S.-X. Chen, A function involving gamma function and having logarithmically absolute convexity, Integral Transforms Spec. Funct., 18(11)(2007), 837-843. Google Scholar
[45]	F. Qi, S. Guo, and B.-N. Guo, Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math., 233(9)(2010), 2149-2160. Google Scholar
[46]	F. Qi, S. Guo, B.-N. Guo, and S.-X. Chen, A class of k-log-convex functions and their applications to some special functions, Integral Transforms Spec. Funct., 19(3)(2008), 195-200. Google Scholar
[47]	F. Qi and 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(2)(2012), 132-158. Google Scholar
[48]	F. Qi and Q.-M. Luo, Complete monotonicity of a function involving the gamma function and applications, Period. Math. Hungar., 69(2)(2014), 159-169. Google Scholar
[49]	F. Qi, Q.-M. Luo, and B.-N. Guo, Complete monotonicity of a function involving the divided difference of digamma functions, Sci. China Math., 56(11)(2013), 2315-2325. Google Scholar
[50]	F. Qi, Q.-M. Luo, and 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 and M. Th. Rassias (Eds), Springer, 2014, 485-494. Google Scholar
[51]	F. Qi, D.-W. Niu, J. Cao, and S.-X. Chen, Four logarithmically completely monotonic functions involving gamma function, J. Korean Math. Soc., 45(2)(2008), 559-573. Google Scholar
[52]	F. Qi and 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(3)(2014), 91-97. Google Scholar
[53]	F. Qi, C.-F. Wei, and B.-N. Guo, Complete monotonicity of a function involving the ratio of gamma functions and applications, Banach J. Math. Anal., 6(1)(2012), 35-44. Google Scholar
[54]	F. Qi and X.-J. Zhang, Complete monotonicity of a difference between the exponential and trigamma functions, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math., 21(2)(2014), 141-145. Google Scholar
[55]	F. Qi, X.-J. Zhang, and W.-H. Li, An integral representation for the weighted geometric mean and its applications, Acta Math. Sin. (Engl. Ser.), 30(1)(2014), 61-68. Google Scholar
[56]	F. Qi, X.-J. Zhang, and W.-H. Li, Lévy-Khintchine representation of the geometric mean of many positive numbers and applications, Math. Inequal. Appl., 17(2)(2014), 719-729. Google Scholar
[57]	F. Qi, X.-J. Zhang, and W.-H. Li, Lévy-Khintchine representations of the weighted geometric mean and the logarithmic mean, Mediterr. J. Math., 11(2)(2014), 315-327. Google Scholar
[58]	Y.-M. Yu, An inequality for ratios of gamma functions, J. Math. Anal. Appl., 352(2)(2009), 967-970. Google Scholar
[59]	T.-H. Zhao, Y.-M. Chu, and Y.-P. Jiang, Monotonic and logarithmically convex properties of a function involving gamma functions, J. Inequal. Appl., 2009(2009), Article ID 728612, 13 pages. Google Scholar
[60]	J.-L. Zhao, B.-N. Guo, and F. Qi, A refinement of a double inequality for the gamma function, Publ. Math. Debrecen, 80(3-4)(2012), 333-342. Google Scholar
[61]	J.-L. Zhao, B.-N. Guo, and F. Qi, Complete monotonicity of two functions involving the tri-and tetra-gamma functions, Period. Math. Hungar., 65(1)(2012), 147-155. Google Scholar