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A completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums

Published online by Cambridge University Press:  17 February 2009

Feng Qi
Feng Qi
Affiliation:
College of Mathematics and Information Science, Henan University, Kaifeng City, Henan Province, 475001, China. Research Institute of Mathematical Inequality Theory, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China; e-mail: qifeng618@gmail.com, qifeng618@hotmail.com
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Abstract

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In this paper, a function involving the divided difference of the psi function is proved to be completely monotonic, a class of inequalities involving sums is found, and an equivalent relation between complete monotonicity and one of the class of inequalities is established.

Keywords

completely monotonic functionpsi functionpolygamma functioninequalitysumequivalence
Type
Research Article
Information
The ANZIAM Journal , Volume 48 , Issue 4 , April 2007 , pp. 523 - 532
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Alzer, H., “Sharp inequalities for the digamma and polygamma functions”, Forum Math. 16 (2004) 181–221.Google Scholar
[2]Batir, N., “Some new inequalities for gamma and polygamma functions”, J. Inequal. Pure Appl. Math. 6 (2005) no. 4, http://jipam.vu.edu.au/article.php?sid=577 Art. 103; Available online at ****. RGMIA Res. Rep. Coll. 7 (2004), no. 3, Art. 1; Available online at http://rgmia.vu.edu.au/v7n3.html.Google Scholar
[3]Berg, C., “Integral representation of some functions related to the gamma function”, Mediterr. J Math. 1 (2004) 433–439.Google Scholar
[4]Chen, Ch.-P., “Monotonicity and convexity for the gamma function”, J. Inequal. Pure Appl. Math. 6 (2005) no. 4, Art. 100; Available online at http://jipam.vu.edu.au/article.php?sid=574.Google Scholar
[5]Elbert, Á. and Laforgia, A., “On some properties of the gamma function”, Proc. Amer. Math. Soc. 128 (2000) 2667–2673.CrossRefGoogle Scholar
[6]Elezović, N., Giordano, C. and Pečarić, J., “Thebest bounds in Gautschi's inequality”, Math. Inequal. Appl. 3 (2000) 239–252.Google Scholar
[7]Gautschi, W., “Some elementary inequalities relating to the gamma and incomplete gamma function”, J. Math. Phys. 38 (1959) 77–81.Google Scholar
[8]Grinshpan, A. Z. and Ismail, M. E. H., “Completely monotonic functions involving the gamma and q-gamma functions”, Proc. Amer. Math. Soc. 134 (2006) 1153–1160.CrossRefGoogle Scholar
[9]Kershaw, D., “Some extensions of W. Gautschi's inequalities for the gamma function”, Math. Comp. 41 (1983) 607–611.Google Scholar
[10]Qi, F., “The best bounds in Kershaw's inequality and two completely monotonic functions”, RGMIA Res. Rep. Coll. 9 (2006) no. 4, Art. 2; Available online at http://rgmia.vu.edu.au/v9n4.html.Google Scholar
[11]Qi, F., “Certain logarithmically N-alternating monotonic functions involving gamma and q-gamma functions”, Nonlinear Funct. Anal. Appl. (2007) accepted. RGMIA Res. Rep. Coll., 8 (2005), 413–422; Available online at http://rgmia.vu.edu.au/v8n3.html.Google Scholar
[12]Qi, F., Cui, R.-Q., Chen, Ch.-P. and Guo, B.-N., “Some completely monotonic functions involving polygamma functions and an application”, J. Math. Anal. Appl. 310 (2005) 303–308.CrossRefGoogle Scholar
[13]Qi, F. and Guo, B.-N., “Complete monotonicities of functions involving the gamma and digamma functions”, RGMIA Res. Rep. Coll. 7 (2004) 63–72; Available online at http://rgmia.vu.edu.au/v7n1.html.Google Scholar
[14]Qi, F., Guo, B.-N. and Chen, Ch.-P., “The best bounds in Gautschi-Kershaw inequalities”, Math. Inequal. Appl. 9 (2006) 427–136. RGMIA Res. Rep. Coll. 8 (2005), no. 2, Art. 17; Available online at http://rgmia.vu.edu.au/v8n2.html.Google Scholar
[15]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. RGMIA Res. Rep. Coll. 7 (2004), 31–36; Available online at http://rgmia.vu.edu.au/v7n1.html.Google Scholar
[16]van Haeringen, H., “Completely monotonic and related functions”, Report 93–108, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands, 1993.Google Scholar
[17]Widder, D. V., The Laplace transform (Princeton University Press, Princeton, 1941)Google Scholar