In the paper, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic. This extends and generalizes the main result of Guo and Qi [Taiwanese J. Math. 7 (2003), no. 2, 239--247] and others. As applications, several inequalities involving the volume of the unit ball in $\mathbb{R}^n$ are derived, which refine, generalize and extend some known inequalities.
References
S. Abramovich, J. Barić, M. Matić and J. Pečarić, On van de Lune-Alzer's inequality, J. Math. Inequal. 1 (2007), no. 4, 563–587. MR2408409 1149.26018 10.7153/jmi-01-47 S. Abramovich, J. Barić, M. Matić and J. Pečarić, On van de Lune-Alzer's inequality, J. Math. Inequal. 1 (2007), no. 4, 563–587. MR2408409 1149.26018 10.7153/jmi-01-47
H. Alzer, Inequalities for the volume of the unit ball in $\mathbb{R}^n$, J. Math. Anal. Appl. 252 (2000), 353–363. MR1797860 0972.26015 10.1006/jmaa.2000.7065 H. Alzer, Inequalities for the volume of the unit ball in $\mathbb{R}^n$, J. Math. Anal. Appl. 252 (2000), 353–363. MR1797860 0972.26015 10.1006/jmaa.2000.7065
H. Alzer, Inequalities for the volume of the unit ball in $\mathbb{R}^n$, I\!I, Mediterr. J. Math. 5 (2008), 395–413. MR2465568 10.1007/s00009-008-0158-x H. Alzer, Inequalities for the volume of the unit ball in $\mathbb{R}^n$, I\!I, Mediterr. J. Math. 5 (2008), 395–413. MR2465568 10.1007/s00009-008-0158-x
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. MR939205 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. MR939205
G. Bennett, Meaningful inequalities, J. Math. Inequal. 1 (2007), no. 4, 449–471. MR2408401 1145.26307 10.7153/jmi-01-39 G. Bennett, Meaningful inequalities, J. Math. Inequal. 1 (2007), no. 4, 449–471. MR2408401 1145.26307 10.7153/jmi-01-39
C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433–439. MR2112748 10.1007/s00009-004-0022-6 1162.33300 C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433–439. MR2112748 10.1007/s00009-004-0022-6 1162.33300
C. Berg and H.L. Pedersen, A one-parameter family of Pick functions defined by the Gamma function and related to the volume of the unit ball in $n$-space, Proc. Amer. Math. Soc. 139 (2011), no. 6, 2121–2132. MR2775390 1230.30021 10.1090/S0002-9939-2010-10636-6 C. Berg and H.L. Pedersen, A one-parameter family of Pick functions defined by the Gamma function and related to the volume of the unit ball in $n$-space, Proc. Amer. Math. Soc. 139 (2011), no. 6, 2121–2132. MR2775390 1230.30021 10.1090/S0002-9939-2010-10636-6
C. Berg and H.L. Pedersen, A Pick function related to the sequence of volumes of the unit ball in $n$-space, Available online at http://arxiv.org/abs/0912.2185. http://arxiv.org/abs/0912.2185 C. Berg and H.L. Pedersen, A Pick function related to the sequence of volumes of the unit ball in $n$-space, Available online at http://arxiv.org/abs/0912.2185. http://arxiv.org/abs/0912.2185
C.-P. Chen, F. Qi, P. Cerone and S.S. Dragomir, Monotonicity of sequences involving convex and concave functions, Math. Inequal. Appl. 6 (2003), no. 2, 229–239. MR1974603 C.-P. Chen, F. Qi, P. Cerone and S.S. Dragomir, Monotonicity of sequences involving convex and concave functions, Math. Inequal. Appl. 6 (2003), no. 2, 229–239. MR1974603
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. MR2196051 1085.33001 10.1090/S0002-9939-05-08050-0 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. MR2196051 1085.33001 10.1090/S0002-9939-05-08050-0
B.-N. Guo, R.-J. Chen and F. Qi, A class of completely monotonic functions involving the polygamma functions, J. Math. Anal. Approx. Theory 1 (2006), no. 2, 124–134. MR2331512 1204.33002 B.-N. Guo, R.-J. Chen and F. Qi, A class of completely monotonic functions involving the polygamma functions, J. Math. Anal. Approx. Theory 1 (2006), no. 2, 124–134. MR2331512 1204.33002
B.-N. Guo and F. Qi, A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications, J. Korean Math. Soc. 48 (2011), no. 3, 655–667. MR2815762 05899089 10.4134/JKMS.2011.48.3.655 B.-N. Guo and F. Qi, A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications, J. Korean Math. Soc. 48 (2011), no. 3, 655–667. MR2815762 05899089 10.4134/JKMS.2011.48.3.655
B.-N. Guo and F. Qi, A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2010), no. 2, 21–30. MR2683163 B.-N. Guo and F. Qi, A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2010), no. 2, 21–30. MR2683163
B.-N. Guo and F. Qi, An extension of an inequality for ratios of gamma functions, J. Approx. Theory 163 (2011), no. 9, 1208–1216. MR2832751 1230.33005 10.1016/j.jat.2011.04.003 B.-N. Guo and F. Qi, An extension of an inequality for ratios of gamma functions, J. Approx. Theory 163 (2011), no. 9, 1208–1216. MR2832751 1230.33005 10.1016/j.jat.2011.04.003
B.-N. Guo and F. Qi, Inequalities and monotonicity for the ratio of gamma functions, Taiwanese J. Math. 7 (2003), no. 2, 239–247. MR1978013 1050.26012 B.-N. Guo and F. Qi, Inequalities and monotonicity for the ratio of gamma functions, Taiwanese J. Math. 7 (2003), no. 2, 239–247. MR1978013 1050.26012
B.-N. Guo and F. Qi, Monotonicity of sequences involving geometric means of positive sequences with monotonicity and logarithmical convexity, Math. Inequal. Appl. 9 (2006), no. 1, 1–9. MR2198544 B.-N. Guo and F. Qi, Monotonicity of sequences involving geometric means of positive sequences with monotonicity and logarithmical convexity, Math. Inequal. Appl. 9 (2006), no. 1, 1–9. MR2198544
B.-N. Guo and F. Qi, Refinements of lower bounds for polygamma functions, Proc. Amer. Math. Soc. (2012), in press. MR3003692 06140726 10.1090/S0002-9939-2012-11387-5 B.-N. Guo and F. Qi, Refinements of lower bounds for polygamma functions, Proc. Amer. Math. Soc. (2012), in press. MR3003692 06140726 10.1090/S0002-9939-2012-11387-5
B.-N. Guo and F. Qi, Some properties of the psi and polygamma functions, Hacet. J. Math. Stat. 39 (2010), 219–231. MR2681248 1203.33002 B.-N. Guo and F. Qi, Some properties of the psi and polygamma functions, Hacet. J. Math. Stat. 39 (2010), 219–231. MR2681248 1203.33002
B.-N. Guo and F. Qi, Two new proofs of the complete monotonicity of a function involving the psi function, Bull. Korean Math. Soc. 47 (2010), no. 1, 103–111. MR2604236 1183.33002 10.4134/BKMS.2010.47.1.103 B.-N. Guo and F. Qi, Two new proofs of the complete monotonicity of a function involving the psi function, Bull. Korean Math. Soc. 47 (2010), no. 1, 103–111. MR2604236 1183.33002 10.4134/BKMS.2010.47.1.103
B.-N. Guo, F. Qi and H.M. Srivastava, Some uniqueness results for the non-trivially complete monotonicity of a class of functions involving the polygamma and related functions, Integral Transforms Spec. Funct. 21 (2010), no. 11, 103–111. MR2739394 10.1080/10652461003748112 1207.26020 B.-N. Guo, F. Qi and H.M. Srivastava, Some uniqueness results for the non-trivially complete monotonicity of a class of functions involving the polygamma and related functions, Integral Transforms Spec. Funct. 21 (2010), no. 11, 103–111. MR2739394 10.1080/10652461003748112 1207.26020
R.A. Horn, On infinitely divisible matrices, kernels and functions, Z. Wahrscheinlichkeitstheorie und Verw. Geb 8 (1967), 219–230. MR217836 10.1007/BF00531524 R.A. Horn, On infinitely divisible matrices, kernels and functions, Z. Wahrscheinlichkeitstheorie und Verw. Geb 8 (1967), 219–230. MR217836 10.1007/BF00531524
F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058, 84 pages. MR2611044 1194.33005 F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058, 84 pages. MR2611044 1194.33005
}.tex-soochow} F. Qi, Inequalities and monotonicity of sequences involving $\sqrt[n]{(n+k)!/k!}\,$, Soochow J. Math. 29 (2003), no. 4, 353–361. MR2020734 1044.05006}.tex-soochow} F. Qi, Inequalities and monotonicity of sequences involving $\sqrt[n]{(n+k)!/k!}\,$, Soochow J. Math. 29 (2003), no. 4, 353–361. MR2020734 1044.05006
F. Qi and C.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), no. 2, 603–607. MR2075188 1046.33001 10.1016/j.jmaa.2004.04.026 F. Qi and C.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), no. 2, 603–607. MR2075188 1046.33001 10.1016/j.jmaa.2004.04.026
F. Qi and B.-N. Guo, A logarithmically completely monotonic function involving the gamma function, Taiwanese J. Math. 14 (2010), no. 4, 1623–1628. MR2663936 1216.26006 F. Qi and B.-N. Guo, A logarithmically completely monotonic function involving the gamma function, Taiwanese J. Math. 14 (2010), no. 4, 1623–1628. MR2663936 1216.26006
F. Qi and B.-N. Guo, An inequality involving the gamma and digamma functions, Available online at http://arxiv.org/abs/1101.4698. http://arxiv.org/abs/1101.4698 F. Qi and B.-N. Guo, An inequality involving the gamma and digamma functions, Available online at http://arxiv.org/abs/1101.4698. http://arxiv.org/abs/1101.4698
F. Qi and B.-N. Guo, Necessary and sufficient conditions for a function involving a ratio of gamma functions to be logarithmically completely monotonic, Available online at http://arxiv.org/abs/0904.1101. http://arxiv.org/abs/0904.1101 F. Qi and B.-N. Guo, Necessary and sufficient conditions for a function involving a ratio of gamma functions to be logarithmically completely monotonic, Available online at http://arxiv.org/abs/0904.1101. http://arxiv.org/abs/0904.1101
F. Qi and B.-N. Guo, Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic, Adv. Appl. Math. 44 (2010), no. 1, 71–83. MR2552656 1179.26036 10.1016/j.aam.2009.03.003 F. Qi and B.-N. Guo, Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic, Adv. Appl. Math. 44 (2010), no. 1, 71–83. MR2552656 1179.26036 10.1016/j.aam.2009.03.003
F. Qi and B.-N. Guo, Some logarithmically completely monotonic functions related to the gamma function, J. Korean Math. Soc. 47 (2010), no. 6, 1283–1297. MR2744772 1208.26017 10.4134/JKMS.2010.47.6.1283 F. Qi and B.-N. Guo, Some logarithmically completely monotonic functions related to the gamma function, J. Korean Math. Soc. 47 (2010), no. 6, 1283–1297. MR2744772 1208.26017 10.4134/JKMS.2010.47.6.1283
F. Qi and B.-N. Guo, Monotonicity and logarithmic convexity relating to the volume of the unit ball, Available online at http://arxiv.org/abs/0902.2509. http://arxiv.org/abs/0902.2509 F. Qi and B.-N. Guo, Monotonicity and logarithmic convexity relating to the volume of the unit ball, Available online at http://arxiv.org/abs/0902.2509. http://arxiv.org/abs/0902.2509
F. Qi, B.-N. Guo and C.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, J. Aust. Math. Soc. 80 (2006), 81–88. MR2212317 1094.33002 10.1017/S1446788700011393 F. Qi, B.-N. Guo and C.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, J. Aust. Math. Soc. 80 (2006), 81–88. MR2212317 1094.33002 10.1017/S1446788700011393
F. Qi, B.-N. Guo and C.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, 31–36. F. Qi, B.-N. Guo and C.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, 31–36.
F. Qi and S. Guo, On a new generalization of Martins' inequality, J. Math. Inequal. 1 (2007), no. 4, 503–514. MR2408404 1151.26018 10.7153/jmi-01-42 F. Qi and S. Guo, On a new generalization of Martins' inequality, J. Math. Inequal. 1 (2007), no. 4, 503–514. MR2408404 1151.26018 10.7153/jmi-01-42
F. Qi, S. Guo and B.-N. Guo, Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math. 233 (2010), no. 9, 2149–2160. MR2577754 1188.26007 10.1016/j.cam.2009.09.044 F. Qi, S. Guo and B.-N. Guo, Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math. 233 (2010), no. 9, 2149–2160. MR2577754 1188.26007 10.1016/j.cam.2009.09.044
F. Qi, W. Li and B.-N. Guo, Generalizations of a theorem of I. Schur, RGMIA Res. Rep. Coll. 9 (2006), no. 3, Art. 15. MR2262711 1156.26301 F. Qi, W. Li and B.-N. Guo, Generalizations of a theorem of I. Schur, RGMIA Res. Rep. Coll. 9 (2006), no. 3, Art. 15. MR2262711 1156.26301
F. Qi and Q.-M. Luo, Generalization of H. Minc and L. Sathre's inequality, Tamkang J. Math. 31 (2000), no. 2, 145–148. MR1762725 0987.26018 F. Qi and Q.-M. Luo, Generalization of H. Minc and L. Sathre's inequality, Tamkang J. Math. 31 (2000), no. 2, 145–148. MR1762725 0987.26018
F. Qi and J.-S. Sun, A monotonicity result of a function involving the gamma function, Anal. Math. 32 (2006), no. 4, 279–282. MR2310106 10.1007/s10476-006-0012-y F. Qi and J.-S. Sun, A monotonicity result of a function involving the gamma function, Anal. Math. 32 (2006), no. 4, 279–282. MR2310106 10.1007/s10476-006-0012-y
R.L. Schilling, R. Song and Z. Vondraček, Bernstein Functions, de Gruyter Studies in Mathematics 37, De Gruyter, Berlin, Germany, 2010. MR2598208 1197.33002 R.L. Schilling, R. Song and Z. Vondraček, Bernstein Functions, de Gruyter Studies in Mathematics 37, De Gruyter, Berlin, Germany, 2010. MR2598208 1197.33002
Y. Yu, An inequality for ratios of gamma functions, J. Math. Anal. Appl. 352 (2009), no. 2, 967–970. MR2501942 1160.33302 10.1016/j.jmaa.2008.11.040 Y. Yu, An inequality for ratios of gamma functions, J. Math. Anal. Appl. 352 (2009), no. 2, 967–970. MR2501942 1160.33302 10.1016/j.jmaa.2008.11.040
T.-H. Zhao, Y.-M. Chu and Y.-P. Jiang, Monotonic and logarithmically convex properties of a function involving gamma functions, J. Inequal. Appl. 2009 (2009), Article ID 728612, 13 pages. MR2496267 1176.33003 T.-H. Zhao, Y.-M. Chu and Y.-P. Jiang, Monotonic and logarithmically convex properties of a function involving gamma functions, J. Inequal. Appl. 2009 (2009), Article ID 728612, 13 pages. MR2496267 1176.33003
