A new lower bound in the second Kershaw's double inequality

Journal of Computational and Applied Mathematics

Volumn 214, Issue 2, 2008, Pages 610-616

  Qi, Feng ^a,b  

^a HENAN UNIVERSITY   (China)

^b HENAN POLYTECHNIC UNIVERSITY   (China)

Author keywords

Comparison; Gamma function; Inequality; Kershaw's double inequality; Logarithmic mean; Logarithmically completely monotonic function; Lower bound; Psi function

Indexed keywords

FUNCTION EVALUATION; PROBLEM SOLVING; THEOREM PROVING;

GAMMA FUNCTION; KERSHAW'S DOUBLE INEQUALITY; LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTION; LOWER BOUND; PSI FUNCTION;

NUMERICAL METHODS;

EID: 39049173057     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2007.03.016     Document Type: Article

References (21)

1. M. Abramowitz, I.A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, vol. 55, 9th printing, Washington, 1970.
2. Batir N. Some new inequalities for gamma and polygamma functions. J. Inequal. Pure Appl. Math. 6 4 (2005). http://jipam.vu.edu.au/article.php?sid=577 Art. 103. Available online at 〈http://jipam.vu.edu.au/article.php?sid=577〉. RGMIA Res. Rep. Coll. 7 (3) (2004) Art. 1. Available online at 〈http://rgmia.vu.edu.au/v7n3.html〉
3. Batir N. Some gamma function inequalities. RGMIA Res. Rep. Colloq. 9 3 (2006). http://rgmia.vu.edu.au/v9n3.html Art. 5. Available online at 〈http://rgmia.vu.edu.au/v9n3.html〉
4. Berg C. Integral representation of some functions related to the gamma function. Mediterr. J. Math. 1 4 (2004) 433-439
5. Elezović N., Giordano C., and Pečarić J. The best bounds in Gautschi's inequality. Math. Inequal. Appl. 3 (2000) 239-252
6. Kershaw D. Some extensions of W. Gautschi's inequalities for the gamma function. Math. Comput. 41 164 (1983) 607-611
7. Qi F. Generalized abstracted mean values. J. Inequal. Pure Appl. Math. 1 1 (2000). http://jipam.vu.edu.au/article.php?sid=97 Art. 4. Available online at 〈http://jipam.vu.edu.au/article.php?sid=97〉. RGMIA Res. Rep. Coll. 2(5) (1999) Art. 4, 633-642. Available online at 〈http://rgmia.vu.edu.au/v2n5.html〉
8. F. Qi, Certain logarithmically N-alternating monotonic functions involving gamma and q-gamma functions, Nonlinear Funct. Anal. Appl. (2007), accepted. RGMIA Res. Rep. Coll. 8 (3) (2005) Art. 5, 413-422. Available online at 〈http://rgmia.vu.edu.au/v8n3.html〉.
9. F. Qi, A class of logarithmically completely monotonic functions and application to the best bounds in the second Gautschi-Kershaw's inequality, RGMIA Res. Rep. Coll. 9 (4) (2006) Art. 11. Available online at 〈http://rgmia.vu.edu.au/v9n4.html〉.
10. F. Qi, A class of logarithmically completely monotonic functions and the best bounds in the first Kershaw's double inequality, J. Comput. Appl. Math. (2007), in press, Available online at 〈http://dx.doi.org/10.1016/j.cam.2006.09.005〉. RGMIA Res. Rep. Coll. 9 (2) (2006) Art. 16. Available online at 〈http://rgmia.vu.edu.au/v9n2.html.
11. x + β, submitted.
12. Qi F., and Chen Ch.-P. A complete monotonicity property of the gamma function. J. Math. Anal. Appl. 296 2 (2004) 603-607
13. Qi F., and Guo B.-N. Complete monotonicities of functions involving the gamma and digamma functions. RGMIA Res. Rep. Coll. 7 1 (2004). http://rgmia.vu.edu.au/v7n1.html Art. 8, 63-72. Available online at 〈http://rgmia.vu.edu.au/v7n1.html〉
14. F. Qi, 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. (2007), in press. Available online at 〈http://dx.doi.org/10.1016/j.cam.2006.12.022〉.
15. F. Qi, B.-N. Guo, Wendel-Gautschi-Kershaw's inequalities and sufficient and necessary conditions that a class of functions involving ratio of gamma functions are logarithmically completely monotonic, RGMIA Res. Rep. Coll. 10 (1) (2007). Available online at 〈http://rgmia.vu.edu.au/v10n1.html〉.
16. Qi F., Guo B.-N., and Chen Ch.-P. Some completely monotonic functions involving the gamma and polygamma functions. RGMIA Res. Rep. Coll. 7 1 (2004). http://rgmia.vu.edu.au/v7n1.html Art. 5, 31-36. Available online at 〈http://rgmia.vu.edu.au/v7n1.html〉
17. Qi F., Guo B.-N., and Chen Ch.-P. The best bounds in Gautschi-Kershaw inequalities. Math. Inequal. Appl. 9 3 (2006) 427-436. http://rgmia.vu.edu.au/v8n2.html RGMIA Res. Rep. Coll. 8 (2) (2005) Art. 17. Available online at 〈http://rgmia.vu.edu.au/v8n2.html〉
18. Qi F., Guo B.-N., and Chen Ch.-P. Some completely monotonic functions involving the gamma and polygamma functions. J. Austral. Math. Soc. 80 (2006) 81-88
19. Qi F., Wei Z.-L., and Yang Q. Generalizations and refinements of Hermite-Hadamard's inequality. Rocky Mountain J. Math. 35 1 (2005) 235-251. http://rgmia.vu.edu.au/v5n2.html RGMIA Res. Rep. Coll. 5 (2) (2002) Art. 10, 337-349. Available online at 〈http://rgmia.vu.edu.au/v5n2.html〉
20. E.W. Weisstein, Faá di Bruno's Formula, From MathWorld-A Wolfram Web Resource. Available online at 〈http://mathworld.wolfram.com/FaadiBrunosFormula.html〉.
21. Widder D.V. The Laplace Transform (1946), Princeton University Press, Princeton