A function involving gamma function and having logarithmically absolute convexity

Integral Transforms and Special Functions

Volumn 18, Issue 11, 2007, Pages 837-843

  Qi, Feng ^a,b   Guo, Bai Ni ^b   Guo, Senlin ^c   Chen, Shou Xin ^a  

^a HENAN UNIVERSITY   (China)

^b HENAN POLYTECHNIC UNIVERSITY   (China)

^c UNIVERSITY OF MANITOBA   (Canada)

Author keywords

Gamma function; Generalization; Logarithmically absolutely convex function; Logarithmically absolutely monotonic function; Logarithmically completely monotonic function

Indexed keywords

EID: 35649001010     PISSN: 10652469     EISSN: 14768291     Source Type: Journal    
DOI: 10.1080/10652460701528875     Document Type: Article

References (38)

1. Alsina, C. and Tomás, M.S., 2005, A geometrical proof of a new inequality for the gamma function. Journal of Inequalities in Pure and Applied Mathematics, 6(2), Art. 48. Available online at http://jipam.vu.edu.au/article.php?sid=517.
2. Sándor, J., 2005, A note on certain inequalities for the gamma function. Journal of Inequalities in Pure and Applied Mathematics, 6(3), Art. 61. Available online at http://jipam.vu.edu.au/article.php? sid=534.
3. Neuman, E., 2006, Inequalities involving a logarithmically convex function and their applications to special functions. Journal of Inequalities in Pure and Applied Mathematics, 7(1), Art. 16. Available online at http://jipam.vu.edu.au/article.php?sid=646.
4. Mercer, A.McD., 2006, Some new inequalities for the gamma, beta and zeta functions. Journal of Inequalities in Pure and Applied Mathematics, 7(1), Art. 29. Available online at http://jipam.vu.edu.au/article.php? sid=636.
5. Mitrinović, D.S., Pečarić, J.E., and Fink, A.M., 1993, Classical and New Inequalities in Analysis (Dordrecht, Boston, London: Kluwer Academic Publishers).
6. van Haeringen, H., 1993, Completely Monotonic and Related Functions, Report 93-108, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands.
7. Widder, D. V., 1941, The Laplace Transform (Princeton: Princeton University Press).
8. Qi, F. and Guo, B.-N., 2004, Complete monotonicities of functions involving the gamma and digamma functions. RGMIA Research Report Collection, 7(1), Art. 8, 63-72. Available online at http://rgmia.vu.edu.au/v7n1.html.
9. Qi, F., Guo, B.-N. and Chen, Ch.-P., 2006, Some completely monotonic functions involving the gamma and polygamma functions. Journal of Australian Mathematical Society, 80(1), 81-88.
10. Qi, F., Guo, B.-N. and Chen, Ch.-P., 2004, Some completely monotonic functions involving the gamma and polygamma functions. RGMIA Research Report Collection, 7(1), Art. 5, 31-36. Available online at http://rgmia.vu.edu.au/v7n1.html.
11. Atanassov, R.D. and Tsoukrovski, U.V., 1988, Some properties of a class of logarithmically completely monotonic functions. Comptes Rendus de l'Académie Bulgare des Sciences, 41(2), 21-23.
12. Berg, C., 2004, Integral representation of some functions related to the gamma function. Mediterranean Journal of Mathematics, 1(4), 433-439.
13. x+β. Integral Transforms and Special Functions, in press.
14. Qi, F. and Chen, Ch.-P., 2004, A complete monotonicity property of the gamma function. Journal of Mathematical Analysis and Application, 296(2), 603-607.
15. Grinshpan, A.Z. and Ismail, M.E.H., 2006, Completely monotonic functions involving the gamma and q-gamma functions. Proceedings of the American Mathematical Society, 134(4), 1153-1160.
16. Horn., R.A., 1967, On infinitely divisible matrices, kernels and functions. Z Wahrscheinlichkeitstheorie und Verw. Geb, 8, 219-230.
17. Chen, Ch.-P. and Qi, F., 2006, Logarithmically completely monotonic functions relating to the gamma function. Journal of Mathematical Analysis and Application, 321(1), 405-411.
18. Chen, Ch.-P. and Qi, F., 2005, Logarithmically completely monotonic ratios of mean values and an application. Global Journal of Mathematics and Mathematical Sciences, 1(1), 71-76.
19. Chen, Ch.-P. and Qi, F., 2005, Logarithmically completely monotonic ratios of mean values and an application. RGMIA Research Report Collection, 8(1), Art. 18. Available online at http://rgmia.vu.edu. au/v8n1.html.
20. Chen, Ch.-P. and Qi, F., 2005, Logarithmically complete monotonicity properties for the gamma functions. Australian Journal of Mathematical Analysis and Application, 2(2), Art. 8. Available online at http://ajmaa.org/cgi-bin/paper.pl?string=v2n2/V2I2P8.tex.
21. Guo, S., 2006, Monotonicity and concavity properties of some functions involving the gamma function with applications. Journal of Inequalities in Pure and Applied Mathematics, 7(2), Art. 45. Available online at http://jipam.vu.edu.au/article.php?sid=662.
22. Guo, S., 2007, On the composition of completely monotonic and related functions. RGMIA Research Report Collection, 10(1), Art. 8. Available online at http://rgmia.vu.edu.au/v10n1.html.
23. Guo, S., 2006, Some function classes connected to the class of completely monotonic functions. RGMIA Research Report Collection, 9(2), Art. 8. Available online at http://rgmia.vu.edu.au/v9n2.html.
24. Qi, F., 2005, Certain logarithmically N-alternating monotonic functions involving gamma and q-gamma functions. RGMIA Research Report Collection, 8(3), Art. 5, 413-422. Available online at http://rgmia.vu.edu.au/v8n3.html.
25. Qi, F., Chen, Sh.-X., and Cheung, W.-S., 2007, Logarithmically completely monotonic functions concerning gamma and digamma functions, Integral Transforms and Special Functions, 18(6), 435-443.
26. Qi, F., Guo, B.-N. and Chen, Ch.-P., 2005, The best bounds in Gautschi-Kershaw inequalities. RGMIA Research Report Collection, 8(2), Art. 17, 311-320. Available online at http://rgmia.vu.edu.au/v8n2. html.
27. Qi, F., Guo, B.-N. and Chen, Ch.-P., 2006, The best bounds in Gautschi-Kershaw inequalities. Mathematical Inequalities and Applications, 9(3), 427-436.
28. Qi, F. and Li, W., 2005, Two logarithmically completely monotonic functions connected with the gamma function. RGMIA Research Report Collection, 8(3), Art. 13, 497-493. Available online at http://rgmia.vu.edu.au/v8n3.html.
29. Qi, F., Yang, Q. and Li, W., 2006, Two logarithmically completely monotonic functions connected with gamma function. Integral Transforms and Special Functions, 17(7), 539-542.
30. Bougoffa, L., 2006, Some inequalities involving the gamma function. Journal of Inequalities in Pure and Applied Mathematics, 7(5), Art. 179. Available online at http://jipam.vu.edu.au/article.php?sid=796.
31. Qi, F., 1998, Generalized weighted mean values with, two parameters. Proceedings of the Royal Society of London Series A. Mathematical, Physical and Engineering Sciences, 454(1978), 2723-2732.
32. x)/=/x:Inequalities and properties. Proceedings of the American Mathematical Society, 126(11), 3355-3359.
33. Qi, F., Li, W. and Guo, B.-N., 2006, Generalizations of a theorem of I. Schur. RGMIA Research Report Collection, 9(3), Art. 15. Available online athttp://rgmia.vu.edu.au/v9n3.html.
34. Qi, F., Li, W. and Guo, B.-N., 2006, Generalizations of a theorem, of I. Schur. Bùdengshì Yânjiû Tôngxùn (Communications in Studies on Inequalities) 13(4), 355-364.
35. Qi, F., 2002, Logarithmic convexity of extended mean values, Proceedings of the American Mathematical Society, 130(6), 1787-1796.
36. Qi, F., 1999, Logarithmic convexity of the extended mean values. RGMIA Research Report Collection, 2(5), Art. 5, 643-652 Available online at http://rgmia.vu.edu.au/v2n5.html.
37. Alzer, H., 2004, Sharp inequalities for the digamma and polygamma functions. Forum Mathematicum, 16 (2004), 1851-221.
38. Qi, F., Guo, S. and Guo, B.-N., 2006, Note on a class of completely monotonic functions involving the polygamma functions. RGMIA Research Report Collection, 10(1), Art. 5. Available online at http://rgmia.vu.edu. au/v10n1.html.