A CLASS OF COMPLETELY MONOTONIC FUNCTIONS INVOLVING DIVIDED DIFFERENCES OF THE PSI AND TRI-GAMMA FUNCTIONS AND SOME APPLICATIONS

- Journal title : Journal of the Korean Mathematical Society
- Volume 48, Issue 3, 2011, pp.655-667
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2011.48.3.655

Title & Authors

A CLASS OF COMPLETELY MONOTONIC FUNCTIONS INVOLVING DIVIDED DIFFERENCES OF THE PSI AND TRI-GAMMA FUNCTIONS AND SOME APPLICATIONS

Guo, Bai-Ni; Qi, Feng;

Guo, Bai-Ni; Qi, Feng;

Abstract

A class of functions involving divided differences of the psi and tri-gamma functions and originating from Kershaw`s double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving the ratio of two gamma functions and originating from the establishment of the best upper and lower bounds in Kershaw`s double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in and respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized.

Keywords

completely monotonic function;divided difference;gamma function;psi function;tri-gamma function;probability integral;error function;double factorial;ratio;volume of unit ball;monotonicity;convexity;inequality;generalization;application;

Language

English

Cited by

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References

1.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 4th printing, with corrections, Washington, 1965.

2.

H. Alzer, Inequalities for the volume of the unit ball in ${\mathbb{R}}^n$ , J. Math. Anal. Appl. 252 (2000), no. 1, 353-363.

3.

H. Alzer, Sharp inequalities for the digamma and polygamma functions, Forum Math. 16 (2004), no. 2, 181-221.

4.

H. Alzer and A. Z. Grinshpan, Inequalities for the gamma and q-gamma functions, J. Approx. Theory 144 (2007), no. 1, 67-83.

5.

R. D. Atanassov and U. V. Tsoukrovski, Some properties of a class of logarithmically completely monotonic functions, C. R. Acad. Bulgare Sci. 41 (1988), no. 2, 21-23.

6.

N. Batir, An interesting double inequality for Euler's gamma function, J. Inequal. Pure Appl. Math. 5 (2004), no. 4, Article 97, 3 pp.; Available online at http://www.emis.de/journals/JIPAM/article452.html?sid=452.

7.

N. Batir, Some new inequalities for gamma and polygamma functions, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Article 103, 9 pp.; Available online at http://www.emis.de/journals/JIPAM/article577.html?sid=577.

8.

C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433-439.

9.

J. Bustoz and M. E. H. Ismail, On gamma function inequalities, Math. Comp. 47 (1986), no. 176, 659-667.

10.

J. Cao, D.-W. Niu, and F. Qi, A Wallis type inequality and a double inequality for probability integral, Aust. J. Math. Anal. Appl. 4 (2007), no. 1, Art. 3, 6 pp.; Available online at http://ajmaa.org/cgi-bin/paper.pl?string=v4n1/V4I1P3.tex.

11.

Ch.-P. Chen, Monotonicity and convexity for the gamma function, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Article 100, 6 pp.; Available online at http://www.emis.de/journals/JIPAM/article457.html?sid=457.

12.

Ch.-P. Chen and F. Qi, Best upper and lower bounds in Wallis' inequality, J. Indones. Math. Soc. 11 (2005), no. 2, 137-141.

13.

Ch.-P. Chen and F. Qi, Completely monotonic function associated with the gamma functions and proof of Wallis' inequality, Tamkang J. Math. 36 (2005), no. 4, 303-307.

14.

Ch.-P. Chen and F. Qi, The best bounds in Wallis' inequality, Proc. Amer. Math. Soc. 133 (2005), no. 2, 397-401.

15.

Ch.-P. Chen and F. Qi, The best bounds to ${\frac{(2n)!}{2^{2n}(n!)^2}}$ , Math. Gaz. 88 (2004), 54-55.

17.

N. Elezovic, C. Giordano and J. Pecaric, The best bounds in Gautschi's inequality, Math. Inequal. Appl. 3 (2000), no. 2, 239-252.

18.

W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys. 38 (1959/60), 77-81.

19.

A. Z. Grinshpan and M. E. H. Ismail, Completely monotonic functions involving the gamma and q-gamma functions, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1153-1160.

20.

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 (2010), no. 2, 21-29.

21.

B.-N. Guo and F. Qi, An alternative proof of Elezovic-Giordano-Pecaric's theorem, Math. Inequal. Appl. 14 (2011), no. 1, in press.

22.

B.-N. Guo and F. Qi, Some logarithmically completely monotonic functions related to the gamma function, J. Korean Math. Soc. 47 (2010), no. 6, 1283-1297; Available online at http://dx.doi.org/10.4134/JKMS.2010.47.6.1283.

23.

D. K. Kazarinoff, On Wallis' formula, Edinburgh Math. Notes 1956 (1956), no. 40, 19-21.

24.

D. Kershaw, Some extensions of W. Gautschi's inequalities for the gamma function, Math. Comp. 41 (1983), no. 164, 607-611.

25.

S. Koumandos, Remarks on a paper by Chao-Ping Chen and Feng Qi, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1365-1367.

26.

27.

I. Lazarevic and A. Lupas, Functional equations for Wallis and gamma functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 461-497 (1974), 245-251.

28.

D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.

29.

F. Qi, A class of logarithmically completely monotonic functions and application to the best bounds in the second Gautschi-Kershaw's inequality, J. Comput. Appl. Math. 224 (2009), no. 2, 538-543; Available online at http://dx.doi.org/10.1016/j.cam.2008.05.030.

30.

F. Qi, A class of logarithmically completely monotonic functions and the best bounds in the first Kershaw's double inequality, J. Comput. Appl. Math. 206 (2007), no. 2, 1007-1014; Available online at http://dx.doi.org/10.1016/j.cam.2006.09.005.

31.

F. Qi, A completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums, ANZIAM J. 48 (2007), no. 4, 523-532.

32.

F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Art. ID 493058, 84 pp.; Available online at http://dx.doi.org/10.1155/2010/493058.

33.

F. Qi, Bounds for the ratio of two gamma functions-From Wendel's limit to Elezovic-Giordano-Pecaric's theorem, Available online at http://arxiv.org/abs/0902.2514.

34.

F. Qi, Monotonicity results and inequalities for the gamma and incomplete gamma functions, Math. Inequal. Appl. 5 (2002), no. 1, 61-67.

35.

F. Qi, Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct. 18 (2007), no. 7-8, 503-509; Available online at http://dx.doi.org/10.1080/10652460701358976.

36.

F. Qi and Ch.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), no. 2, 603-607; Available online at http://dx.doi.org/10.1080/10652460701318418.

37.

F. Qi, L.-H. Cui, and S.-L. Xu, Some inequalities constructed by Tchebysheff's integral inequality, Math. Inequal. Appl. 2 (1999), no. 4, 517-528.

38.

F. Qi and B.-N. Guo, A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw's double inequality, J. Comput. Appl. Math. 212 (2008), no. 2, 444-456; Available online at http://dx.doi.org/10.1016/j.cam.2006.12.022.

39.

F. Qi and B.-N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 8, 63-72; Available online at http://rgmia.org/v7n1.php.

40.

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 (2009), no. 6, 1975-1989; Available online at http://dx.doi.org/10.3934/cpaa.2009.8.1975.

41.

F. Qi and B.-N. Guo, Wendel's and Gautschi's inequalities: refinements, extensions, and a class of logarithmically completely monotonic functions, Appl. Math. Comput. 205 (2008), no. 1, 281-290; Available online at http://dx.doi.org/10.1016/j.amc.2008.07.005.

42.

F. Qi, B.-N. Guo, and Ch.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 5, 31-36; Available online at http://rgmia.org/v7n1.php.

43.

F. Qi, B.-N. Guo, and Ch.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, J. Aust. Math. Soc. 80 (2006), no. 1, 81-88.

44.

F. Qi, B.-N. Guo, and Ch.-P. Chen, The best bounds in Gautschi-Kershaw inequalities, Math. Inequal. Appl. 9 (2006), no. 3, 427-436.

45.

G. N. Watson, A note on Gamma functions, Proc. Edinburgh Math. Soc. (2) 11 (1958/1959).

46.

G. N. Watson, A note on Gamma functions, Edinburgh Math. Notes No. 42 (misprinted 41) (1959), 7-9.

47.

E. W. Weisstein, Wallis Cosine Formula, From MathWorld-A Wolfram Web Resource; Available online at http://mathworld.wolfram.com/WallisCosineFormula.html.

49.

D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946.

50.

Y.-Q. Zhao and Q.-B. Wu, Wallis inequality with a parameter, J. Inequal. Pure Appl. Math. 7 (2006), no. 2, Article 56, 10 pp.; Available online at http://www.emis.de/journals/JIPAM/article673.html?sid=673.