Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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SOME COMPLETELY MONOTONIC FUNCTIONS INVOLVING THE GAMMA AND POLYGAMMA FUNCTIONS

  • Li, Ai-Jun (Department of Mathematics Shanghai University) ;
  • Chen, Chao-Ping (School of Mathematics and Informatics Research Institute of Applied Mathematics Henan Polytechnic University)
  • Published : 2008.01.31
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Abstract

In this paper, some logarithmically completely monotonic, strongly completely monotonic and completely monotonic functions related to the gamma, digamma and polygamma functions are established. Several inequalities, whose bounds are best possible, are obtained.

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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, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), no. 217, 373-389 https://doi.org/10.1090/S0025-5718-97-00807-7
  3. H. Alzer, Inequalities for the volume of the unit ball in Rn, J. Math. Anal. Appl. 252 (2000), no. 1, 353-363 https://doi.org/10.1006/jmaa.2000.7065
  4. H. Alzer, A power mean inequality for the gamma function, Monatsh. Math. 131 (2000), no. 3, 179-188 https://doi.org/10.1007/s006050070007
  5. H. Alzer, Mean-value inequalities for the polygamma functions, Aequationes Math. 61 (2001), no. 1-2, 151-161 https://doi.org/10.1007/s000100050167
  6. H. Alzer, Sharp inequalities for the digamma and polygamma functions, Forum Math. 16 (2004), no. 2, 181-221 https://doi.org/10.1515/form.2004.009
  7. H. Alzer and C. Berg, Some classes of completely monotonic functions. II., Ramanujan J. 11 (2006), no. 2, 225-248 https://doi.org/10.1007/s11139-006-6510-5
  8. H. Alzer and C. Berg, Some classes of completely monotonic functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 445-460
  9. H. Alzer and J. Wells, Inequalities for the polygamma functions, SIAM J. Math. Anal. 29 (1998), no. 6, 1459-1466 https://doi.org/10.1137/S0036141097325071
  10. G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1713-1723 https://doi.org/10.2307/2154966
  11. G. D. Anderson and S.-L. Qiu, A monotoneity property of the gamma function, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3355-3362 https://doi.org/10.1090/S0002-9939-97-04152-X
  12. K. Ball, Completely monotonic rational functions and Hall's marriage theorem, J. Combin. Theory Ser. B 61 (1994), no. 1, 118-124 https://doi.org/10.1006/jctb.1994.1037
  13. C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433-439 https://doi.org/10.1007/s00009-004-0022-6
  14. C. Berg and H. L. Pedersen, A completely monotone function related to the gamma function, J. Comput. Appl. Math. 133 (2001), no. 1-2, 219-230 https://doi.org/10.1016/S0377-0427(00)00644-0
  15. C. Berg and H. L. Pedersen, Pick functions related to the gamma function, Rocky Mountain J. Math. 32 (2002), no. 2, 507-525 https://doi.org/10.1216/rmjm/1030539684
  16. K. H. Borgwardt, The simplex method, A probabilistic analysis. Algorithms and Combinatorics: Study and Research Texts, 1. Springer-Verlag, Berlin, 1987
  17. A. M. Bruckner and E. Ostrow, Some function classes related to the class of convex functions, Pacific J. Math. 12 (1962), 1203-1215 https://doi.org/10.2140/pjm.1962.12.1203
  18. J. Bustoz and M. E. H. Ismail, On gamma function inequalities, Math. Comp. 47 (1986), no. 176, 659-667 https://doi.org/10.2307/2008180
  19. Chao-Ping Chen, Complete monotonicity properties for a ratio of gamma functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 16 (2005), 26-28
  20. P. J. Davis, Leonhard Euler's integral: A historical profile of the gamma function, Amer. Math. Monthly 66 (1959), 849-869 https://doi.org/10.2307/2309786
  21. B. J. English and G. Rousseau, Bounds for certain harmonic sums, J. Math. Anal. Appl. 206 (1997), no. 2, 428-441 https://doi.org/10.1006/jmaa.1997.5226
  22. 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 https://doi.org/10.1090/S0002-9939-05-08050-0
  23. R. A. Horn, On infinitely divisible matrices, kernels, and functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 219-230 https://doi.org/10.1007/BF00531524
  24. D. Kershaw, Some extensions of W. Gautschi's inequalities for the gamma function, Math. Comp. 41 (1983), no. 164, 607-611 https://doi.org/10.2307/2007697
  25. C. H. Kimberling, A probabilistic interpretation of complete monotonicity, Aequationes Math. 10 (1974), 152-164 https://doi.org/10.1007/BF01832852
  26. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52 Springer-Verlag New York, Inc., New York 1966
  27. M. Merkle, On log-convexity of a ratio of gamma functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 8 (1997), 114-119
  28. M. Merkle, Convexity, Schur-convexity and bounds for the gamma function involving the digamma function, Rocky Mountain J. Math. 28 (1998), no. 3, 1053-1066 https://doi.org/10.1216/rmjm/1181071755
  29. M. Merkle, Gurland's ratio for the gamma function, Comput. Math. Appl. 49 (2005), no. 2-3, 389-406 https://doi.org/10.1016/j.camwa.2004.01.016
  30. H. Minc and L. Sathre, Some inequalities involving $(r!)^{1/r}$, Proc. Edinburgh Math. Soc. (2) 14 (1964/1965), 41-46
  31. F. Qi and Ch.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), no. 2, 603-607 https://doi.org/10.1016/j.jmaa.2004.04.026
  32. F. Qi and B.-N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (2004), 63-72, Art. 8
  33. 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 https://doi.org/10.1017/S1446788700011393
  34. S.-L. Qiu and M. Vuorinen, Some properties of the gamma and psi functions, with applications, Math. Comp. 74 (2005), no. 250, 723-742 https://doi.org/10.1090/S0025-5718-04-01675-8
  35. S. Y. Trimble, J. Wells, and F. T. Wright, Superadditive functions and a statistical application, SIAM J. Math. Anal. 20 (1989), no. 5, 1255-1259 https://doi.org/10.1137/0520082
  36. Zh.-X. Wang and D.-R. Guo, T`eshu Hanshu Gailun (Introduction to Special Function), The Series of Advanced Physics of Peking University, Peking University Press, Beijing, China, 2000
  37. D. V. Widder, The Laplace Transform, Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J., 1941

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