A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function

UPB Scientific Bulletin, Series A: Applied Mathematics and Physics

Volumn 72, Issue 2, 2010, Pages 21-30

  Guo, Bai Ni ^a   Qi, Feng ^b  

^a HENAN POLYTECHNIC UNIVERSITY   (China)

^b TIANJIN POLYTECHNIC UNIVERSITY   (China)

Author keywords

Absolutely monotonic function; Completely monotonic function; Di bruno's formula; Fa ; Logarithmically absolutely monotonic function; Logarithmically completely monotonic function; Open problem; Property

Indexed keywords

COMPLETELY MONOTONIC FUNCTION; LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTION; MONOTONIC FUNCTIONS; MONOTONICITY; OPEN PROBLEMS;

EXPONENTIAL FUNCTIONS;

EID: 77952574791     PISSN: 12237027     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (50)

1. H. Alzer, On some inequalities for the Gamma and psi functions, Math. Comp. 66 (1997), no. 217, 373-389.
2. H. Alzer, Sharp inequalities for the digamma and polygamma functions, Forum. Math. 16 (2004), 181-221.
3. H. Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), no. 201, 337-346.
4. H. Alzer and C. Berg, Some classes of completely functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 445-460.
5. G. D. Anderson and S.-L. Qiu, A monotonicity property of the gamma function, Proc. Amer. Math. Soc. 125 (1997), 3355-3362.
6. 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.
7. C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433-439.
8. C. Berg and H. L. Pedersen, A completely monotone function related to the gamma function, Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999). J. Comput. Appl. Math. 133 (2001), no. 1-2, 219-230.
9. J. Bustoz and M. E. H. Ismail, On gamma function inequalities, Math. Comp. 47 (1986), 659-667.
10. Ch.-P. Chen and F. Qi, Logarithmically completely monotonic functions relating to the gamma function, J. Math. Anal. Appl. 321 (2006), no. 1, 405-411.
11. Ch.-P. Chen and F. Qi, Logarithmically complete monotonicity properties for the gamma functions, Austral. J. Math. Anal. Appl. 2 (2005), no. 2, Art. 8; Available online at http: //ajmaa.org/cgi-bin/paper.p1?string=v2n2/V2I2P8.tex.
12. Ch.-P. Chen and F. Qi, Logarithmically completely monotonic ratios of mean values and an application, Glob. J. Math. Math. Sci. 1 (2005), no. 1, 71-76.
13. W. E. Clark and M. E. H. Ismail, Inequalities involving gamma and psi functions, Anal. Appl. (Singap.) 1 (2003), no. 1, 129-140.
14. Á. Elbert and A. Laforgia, On some properties of the gamma function, Proc. Amer. Math. Soc. 128 (2000), no. 9, 2667-2673.
15. C. F. Faá di Bruno, Note sur une nouvelle formule de calcul differentiel, Quarterly J. Pure Appl. Math. 1 (1857), 359-360
16. A. Z. Grinshpan and M. E. H. Ismail, Completely monotonic functions involving the gamma and q-gamma functions, Proc. Amer. Math. Soc. 134 (2006), 1153-1160.
17. B.-N. Guo, On generalization of I. Schur's theorem, Gōngkē Shùxué (Journal of Matematics for Technology) 12 (1996), no. 1, 151-154. (Chinese)
18. B.-N. Guo, On a generalization and proof of I. Schur's problem, Jiāozuò Gōngxuéyuàn Xuébào (Journal of Jiaozuo Institute of Technology) 15 (1996), no. 1, 85-88. (Chinese)
19. n+α, Gāojiào Yánjiū (Jiāozuò Kuàngyè Xuéyuàn) (Studies on Higher Education (Jiaozuo Mining Institute)) (1993), no. 2, 49-51. (Chinese)
20. B.-N. Guo and F. Qi, Some logarithmically completely monotonic functions related to the gamma function, J. Korean Math. Soc. 47 (2010), no. 6, in press.
21. R. A. Horn, On infinitely divisible matrices, kernels and functions, Z. Wahrscheinlichkeitstheorie und Verw. Geb 8 (1967), 219-230.
22. M. E. H. Ismail, L. Lorch, and M. E. Muldoon, Completely monotonic functions associated with the gamma function and its q-analogues, J. Math. Anal. Appl. 116 (1986), 1-9.
23. n, Shùxué Tōngxùn (Communications in Mathematics) (1983), no. 2, 33-34. (Chinese)
24. W. P. Johnson, The curious history of Faá di Bruno's formula, Amer. Math. Monthly 109 (2002), no. 3, 217-234.
25. R. F. Johnsonbaugh, Another proof of an estimate for e, Amer. Math. Monthly 81 (1974), no. 9, 1011-1012.
26. J.-Ch. Kuang, Chángyòng Bùděngshì (Applied Inequalities), 2nd ed., Húnán Jiàoyù Chūbǎn Shè (Hunan Education Press), Changsha City, Hunan Province, China, 1993. (Chinese)
27. J.-Ch. Kuang, Chángyòng Bùdēngshì (Applied Inequalities), 3rd ed., Shāndōng Kēxué Jìshù Chūbǎn Shè (Shandong Science and Technology Press), Ji'nan City, Shandong Province, China, 2004. (Chinese)
28. A.-J. Li, W.-Zh. Zhao and Ch.-P. Chen, Logarithmically complete monotonicity and Schurconvexity for some ratios of gamma functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 17 (2006), 88-92.
29. M. E. Muldoon, Some monotonicity properties and characterizations of the gamma function, Aequationes Math. 18 (1978), 54-63.
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, Certain logarithmically N-alternating monotonic functions involving gamma and q-gamma functions, Nonlinear Funct. Anal. Appl. 12 (2007), no. 4, 675-685.
32. F. Qi, Further generalization of I. Schur's theorem, Jiāozuò Kuàngyè Xuéyuàn Xuébào (Journal of Jiaozuo Mining Institute) 14 (1995), no. 9, 120-124. (Chinese)
33. F. Qi, Generalized weighted mean values with two parameters, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998), no. 1978, 2723-2732.
34. F. Qi, J. Cao, D.-W. Niu, and Sh.-X. Chen, Four logarithmically completely monotonic functions involving gamma function, J. Korean Math. Soc. 45 (2008), no. 2, 559-573.
35. F. Qi and Ch.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), no. 2, 603-607.
36. F. Qi and Ch.-P. Chen, Monotonicities of two sequences, Mathematics and Informatics Quarterly 9 (1999), no. 4, 136-139.
37. F. Qi, Sh.-X. Chen, and W.-S. Cheung, Logarithmically completely monotonic functions concerning gamma and digamma functions, Integral Transforms Spec. Funct. 18 (2007), no. 6, 435-443; Available online at http://dx.doi.org/ 10.1080/10652460701318418.
38. 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.
39. n, Méitàn Gāoděng Jiàoyù (Coal Higher Education), Supplement, November/1993, 32-33. (Chinese)
40. 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.
41. 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), 81-88.
42. F. Qi, W. Li and B.-N. Guo, Generalizations of a theorem of I. Schur, Appl. Math. E-Notes 6 (2006), Art. 29, 244-250.
43. F. Qi, D.-W. Niu, and J. Cao, Logarithmically completely monotonic functions involving gamma and polygamma functions, J. Math. Anal. Approx. Theory 1 (2006), no. 1, 66-74.
44. F. Qi, Q. Yang and W. Li, Two logarithmically completely monotonic functions connected with gamma function, Integral Transforms Spec. Funct. 17 (2006), no. 7, 539-542.
45. x)/x: Inequalities and properties, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3355-3359.
46. H. van Haeringen, Completely Monotonic and Related Functions, Report 93-108, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands, 1993.
47. E. W. Weisstein, Faá di Bruno's Formula, From Math World-A Wolfram Web Resource. http ://mathworld.wolfram.com/FaadiBrunosFormula.html.
48. D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946.
49. X.-Q. Xu, Generalizations and proofs of three results of I. Schur, Shùxué de Shíjiàn yü Rènshí (Mathematics in Practice and Theory) 23 (1993), no. 4, 78-79 and 91. (Chinese)
50. S.-L. Zhang, Ch.-P. Chen and F. Qi, On a completely monotonic function, Shùxué de Shíjiàn yü Rènshí (Mathematics in Practice and Theory) 36 (2006), no. 6, 236-238. (Chinese)