Sharp Inequalities for Polygamma Functions

References

[1] ABRAMOWITZ, M.-STEGUN, I. A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Applied Mathematics Series 55, U.S. Government Printing Office, Washington, 1965.10.1115/1.3625776Search in Google Scholar

[2] ALLASIA, G.-GIORDANO, C.-PEČARIĆ, J.: Inequalities for the gamma function relating to asymptotic expasions, Math. Inequal. Appl. 5 (2002), 543-555.Search in Google Scholar

[3] ALZER, H.: On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), 373-389.10.1090/S0025-5718-97-00807-7Search in Google Scholar

[4] ALZER, H.: Sharp inequalities for the digamma and polygamma functions, ForumMath. 16 (2004), 181-221.Search in Google Scholar

[5] ALZER, H.- BATIR, N.: Monotonicity properties of the gamma function, Appl. Math. Lett. 20 (2007), 778-781; http://dx.doi.org/10.1016/j.aml.2006.08.026.10.1016/j.aml.2006.08.026Search in Google Scholar

[6] ALZER, H.-GRINSHPAN, A. Z.: Inequalities for the gamma and q-gamma functions, J. Approx. Theory 144 (2007), 67-83.10.1016/j.jat.2006.04.008Search in Google Scholar

[7] ATANASSOV, R. D.-TSOUKROVSKI, U. V.: Some properties of a class of logarithmically completely monotonic functions, C. R. Acad. Bulgare Sci. 41 (1988), No. 2, 21-23.Search in Google Scholar

[8] BATIR, N.: An interesting double inequality for Euler’s gamma function, J. Inequal Pure Appl. Math. 5 (2004), No. 4, Art. 97; http://www.emis.de/journals/JIPAM/article452.html.Search in Google Scholar

[9] BATIR, N.: Inequalities for the gamma function, Arch.Math. (Basel) 91 (2008), 554-563.10.1007/s00013-008-2856-9Search in Google Scholar

[10] BATIR, N.: On some properties of digamma and polygamma functions, J. Math. Anal. Appl. 328 (2007), 452-465; http://dx.doi.org/10.1016/j.jmaa.2006.05.065.10.1016/j.jmaa.2006.05.065Search in Google Scholar

[11] BATIR, N.: Some new inequalities for gamma and polygamma functions, J. Inequal. Pure Appl. Math. 6 (2005), No. 4, Art. 103; http://www.emis.de/journals/JIPAM/article577.html.Search in Google Scholar

[12] BERG, C.: Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), 433-439.10.1007/s00009-004-0022-6Search in Google Scholar

[13] CHEN, C.-P.-QI, F.: Logarithmically completely monotonic functions relating to the gamma function, J. Math. Anal. Appl. 321 (2006), no. 1, 405-411; http://dx.doi.org/10.1016/j.jmaa.2005.08.056.10.1016/j.jmaa.2005.08.056Search in Google Scholar

[14] CHU, Y.-M.-ZHANG, X.-M.-TANG, X.-M.: An elementary inequality for psi function, Bull. Inst. Math. Acad. Sin. (N.S.) 3 (2008), 373-380.Search in Google Scholar

[15] CHU, Y.-M.-ZHANG, X.-M.-ZHANG, Z.-H.: The geometric convexity of a function involving gamma function with applications, Commun. Korean Math. Soc. 25 (2010), 373-383; <http://dx.doi.org/10.4134/CKMS.2010.25.3.373>.Search in Google Scholar

[16] ELEZOVIĆ, N.-GIORDANO, C.-PEˇCARIĆ, J.: The best bounds in Gautschi’s inequality, Math. Inequal. Appl. 3 (2000), 239-252.Search in Google Scholar

[17] GRINSHPAN, A. Z.-ISMAIL, M. E. H.: Completely monotonic functions involving the gamma and q-gamma functions, Proc. Amer. Math. Soc. 134 (2006), 1153-1160.10.1090/S0002-9939-05-08050-0Search in Google Scholar

[18] GUO, B.-N.-QI, F.: 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), 655-667; http://dx.doi.org/10.4134/JKMS.2011.48.3.655.10.4134/JKMS.2011.48.3.655Search in Google Scholar

[19] GUO, B.-N.-CHEN, R.-J.-QI, F.: A class of completely monotonic functions involving the polygamma functions, J. Math. Anal. Approx. Theory 1 (2006), 124-134.Search in Google Scholar

[20] GUO, B.-N.-QI, F.: A completely monotonic function involving the tri-gamma function and with degree one, Appl. Math. Comput. 218 (2012), 9890-9897; http://dx.doi.org/10.1016/j.amc.2012.03.075.10.1016/j.amc.2012.03.075Search in Google Scholar

[21] GUO, B.-N.-QI, F.: 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), 21-30.Search in Google Scholar

[22] GUO, B.-N.-QI, F.: An extension of an inequality for ratios of gamma functions, J. Approx. Theory 163 (2011), 1208-1216; http://dx.doi.org/10.1016/j.jat.2011.04.003.10.1016/j.jat.2011.04.003Search in Google Scholar

[23] GUO, B.-N.-QI, F.: Monotonicity of functions connected with the gamma function and the volume of the unit ball, Integral Transforms Spec. Funct. 23 (2012), 701-708; http://dx.doi.org/10.1080/10652469.2011.627511.10.1080/10652469.2011.627511Search in Google Scholar

[24] GUO, B.-N.-QI, F.: Refinements of lower bounds for polygamma functions, Proc. Amer. Math. Soc. 141 (2013), 1007-1015; http://dx.doi.org/10.1090/S0002-9939-2012-11387-5.10.1090/S0002-9939-2012-11387-5Search in Google Scholar

[25] GUO, B.-N.-QI, F.: Some properties of the psi and polygamma functions, Hacet. J. Math. Stat. 39 (2010), 219-231.Search in Google Scholar

[26] GUO, B.-N.-QI, F.: Two new proofs of the complete monotonicity of a function involving the psi function, Bull. Korean Math. Soc. 47 (2010), 103-111; http://dx.doi.org/10.4134/bkms.2010.47.1.103.10.4134/BKMS.2010.47.1.103Search in Google Scholar

[27] GUO, B.-N.-QI, F.-SRIVASTAVA, H. M.: Some uniqueness results for the nontrivially complete monotonicity of a class of functions involving the polygamma and related functions, Integral Transforms Spec. Funct. 21 (2010), 849-858; http://dx.doi.org/10.1080/10652461003748112.10.1080/10652461003748112Search in Google Scholar

[28] GUO, B.-N.-ZHANG, Y.-J.-QI, F.: Refinements and sharpenings of some double inequalities for bounding the gamma function, J. Inequal. Pure Appl. Math. 9 (2008), No. 1, Art. 17; http://www.emis.de/journals/JIPAM/article953.html.Search in Google Scholar

[29] GUO, B.-N.-ZHAO, J.-L.-QI, F.: A completely monotonic function involving the triand tetra-gamma functions, Math. Slovaca 63 (2013), 469-478; http://dx.doi.org/10.2478/s12175-013-0109-2.10.2478/s12175-013-0109-2Search in Google Scholar

[30] GUO, S.-QI, F.-SRIVASTAVA, H. M.: A class of logarithmically completely monotonic functions related to the gamma function with applications, Integral Transforms Spec. Funct. 23 (2012), 557-566; http://dx.doi.org/10.1080/10652469.2011.611331.10.1080/10652469.2011.611331Search in Google Scholar

[31] GUO, S.-QI, F.-SRIVASTAVA, H. M.: Necessary and sufficient conditions for two classes of functions to be logarithmically completely monotonic, Integral Transforms Spec. Funct. 18 (2007), 819-826; http://dx.doi.org/10.1080/10652460701528933.10.1080/10652460701528933Search in Google Scholar

[32] GUO, S.-QI, F.-SRIVASTAVA, H. M.: Supplements to a class of logarithmically completely monotonic functions associated with the gamma function, Appl. Math. Comput. 197 (2008), 768-774; http://dx.doi.org/10.1016/j.amc.2007.08.011.10.1016/j.amc.2007.08.011Search in Google Scholar

[33] GUO, S.-SRIVASTAVA, H. M.: A class of logarithmically completely monotonic functions, Appl. Math. Lett. 21 (2008), 1134-1141. 11710.1016/j.aml.2007.10.028Search in Google Scholar

[34] HORN, R. A.: On infinitely divisible matrices, kernels and functions, Z. Wahrscheinlichkeitstheorie Verw. Geb. 8 (1967), 219-230.10.1007/BF00531524Search in Google Scholar

[35] ISMAIL, M. E. H.-MULDOON, M. E.: Inequalities and monotonicity properties for gamma and q-gamma functions. In: Approximation and Computation: A Festschrift in Honour of Walter Gautschi (R. V. M. Zahar, ed.), Internat. Ser. Numer. Math. 119, Birkhauser, Basel, 1994, pp. 309-323.Search in Google Scholar

[36] KAZARINOFF, D. K.: On Wallis’ formula, Edinburgh Math. Notes 40 (1956), 19-21.10.1017/S095018430000029XSearch in Google Scholar

[37] KOUMANDOS, S.: Remarks on some completely monotonic functions, J. Math. Anal. Appl. 324 (2006), 1458-1461.10.1016/j.jmaa.2005.12.017Search in Google Scholar

[38] KOUMANDOS, S.-PEDERSEN, H. L.: Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function, J. Math. Anal. Appl. 355 (2009), 33-40.10.1016/j.jmaa.2009.01.042Search in Google Scholar

[39] LÜ, Y.-P.-SUN, T.-C.-CHU, T.-C.: Necessary and sufficient conditions for a class of functions and their reciprocals to be logarithmically completely monotonic, J. Inequal. Appl. 2011 (2011), Article 36; http://dx.doi.org/10.1186/1029-242X-2011-36.10.1186/1029-242X-2011-36Search in Google Scholar

[40] MERKLE, M.: Logarithmic convexity and inequalities for the gamma function, J. Math. Anal. Appl. 203 (1996), 369-380.10.1006/jmaa.1996.0385Search in Google Scholar

[41] MITRINOVIĆ, D. S.-PEˇCARIĆ, J. E.-FINK, A. M.: Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993.10.1007/978-94-017-1043-5_17Search in Google Scholar

[42] MORTICI, C.: Very accurate estimates of the polygamma functions, Asymptot. Anal. 68 (2010), 125-134; http://dx.doi.org/10.3233/ASY-2010-0983.10.3233/ASY-2010-0983Search in Google Scholar

[43] MULDOON, M. E.: Some monotonicity properties and characterizations of the gamma function, Aequationes Math. 18 (1978), 54-63.10.1007/BF01844067Search in Google Scholar

[44] QI, F.: A completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums, ANZIAM J. 48 (2007), 523-532; http://dx.doi.org/10.1017/S1446181100003199.10.1017/S1446181100003199Search in Google Scholar

[45] QI, F.: Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058, 84 pp.; http://dx.doi.org/10.1155/2010/493058.10.1155/2010/493058Search in Google Scholar

[46] FENG Qi-QIU-MING LUO: Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to ElezoviĆ-Giordano-Pečari Ć’s theorem, J. Inequal. Appl. 2013 (2013), 542, 20 pages; http://dx.doi.org/10.1186/1029-242X-2013-542.10.1186/1029-242X-2013-542Search in Google Scholar

[47] QI, F.: Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct. 18 (2007), 503-509; http://dx.doi.org/10.1080/10652460701358976.10.1080/10652460701358976Search in Google Scholar

[48] QI, F.-CHEN, C.-P.: A complete monotonicity property of the gamma function, J.Math. Anal. Appl. 296 (2004), 603-607; http://dx.doi.org/10.1016/j.jmaa.2004.04.026.10.1016/j.jmaa.2004.04.026Search in Google Scholar

[49] QI, F.-CUI, R.-Q.-CHEN, C.-P.-GUO, B.-N.: Some completely monotonic functions involving polygamma functions and an application, J. Math. Anal. Appl. 310 (2005), 303-308; http://dx.doi.org/10.1016/j.jmaa.2005.02.016.10.1016/j.jmaa.2005.02.016Search in Google Scholar

[50] QI, F.-GUO, B.-N.: A logarithmically completely monotonic function involving the gamma function, Taiwanese J. Math. 14 (2010), 1623-1628.10.11650/twjm/1500405972Search in Google Scholar

[51] QI, F.-GUO, B.-N.: An inequality involving the gamma and digamma functions, http://arxiv.org/abs/1101.4698.Search in Google Scholar

[52] QI, F.-GUO, B.-N.: Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (2004), No. 1, Art. 8, 63-72; http://rgmia.org/v7n1.php.Search in Google Scholar

[53] QI, F.-GUO, B.-N.: Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications, Commun. Pure Appl. Anal. 8 (2009), 1975-1989; <http://dx.doi.org/10.3934/cpaa.2009.8.1975>.Search in Google Scholar

[54] QI, F.-GUO, B.-N.: Necessary and sufficient conditions for a function involving divided differences of the di- and tri-gamma functions to be completely monotonic, http://arxiv.org/abs/0903.3071.Search in Google Scholar

[55] QI, F.-GUO, B.-N.: Necessary and sufficient conditions for functions involving the triand tetra-gamma functions to be completely monotonic, Adv. Appl. Math. 44 (2010), 71-83; http://dx.doi.org/10.1016/j.aam.2009.03.003.10.1016/j.aam.2009.03.003Search in Google Scholar

[56] QI, F.-GUO, B.-N.: Sharp inequalities for polygamma functions, http://arxiv.org/abs/0903.1984.Search in Google Scholar

[57] GUO, B.-N.-QI, F.: Sharp inequalities for the psi function and harmonic numbers, Analysis (Munich) 34 (2014), 201-208; http://dx.doi.org/10.1515/anly-2014-0001.10.1515/anly-2014-0001Search in Google Scholar

[58] QI, F.-GUO, B.-N.: Some properties of extended remainder of Binet’s first formula for logarithm of gamma function, Math. Slovaca 60 (2010), 461-470; http://dx.doi.org/10.2478/s12175-010-0025-7.10.2478/s12175-010-0025-7Search in Google Scholar

[59] QI, F.-GUO, B.-N.-CHEN, C.-P.: Some completely monotonic functions involving the gamma and polygamma functions, J. Aust. Math. Soc. 80 (2006), 81-88; http://dx.doi.org/10.1017/S1446788700011393.10.1017/S1446788700011393Search in Google Scholar

[60] QI, F.-Guo, S.-GUO, B.-N.: Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math. 233 (2010), 2149-2160; http://dx.doi.org/10.1016/j.cam.2009.09.044.10.1016/j.cam.2009.09.044Search in Google Scholar

[61] QI, F.-LUO, Q.-M.: Bounds for the ratio of two gamma functions-from Wendel’s and related inequalities to logarithmically completely monotonic functions, Banach J. Math. Anal. 6 (2012), 132-158.10.15352/bjma/1342210165Search in Google Scholar

[62] QI, F.-WEI, C.-F.-GUO, B.-N.: Complete monotonicity of a function involving the ratio of gamma functions and applications, Banach J. Math. Anal. 6 (2012), 35-44.10.15352/bjma/1337014663Search in Google Scholar

[63] SONG, Y.-Q.-CHU, Y.-M.-WU, L.-L.: An elementary double inequality for gamma function, Int. J. Pure Appl. Math. 38 (2007), 549-554.Search in Google Scholar

[64] WU, L.-L.-CHU, Y.-M.: An inequality for the psi functions, Appl. Math. Sci. (Ruse) 2 (2008), 545-550.Search in Google Scholar

[65] WU, L.-L.-CHU, Y.-M.-TANG, X.-M.: Inequalities for the generalized logarithmic mean and psi functions, Int. J. Pure Appl. Math. 48 (2008), 117-122.Search in Google Scholar

[66] ZHANG, X.-M.-CHU, Y.-M.: A double inequality for gamma function, J. Inequal. Appl. 2009 (2009), Article ID 503782, 7 pp.; http://dx.doi.org/10.1155/2009/503782.10.1155/2009/503782Search in Google Scholar

[67] ZHANG, X.-M.-CHU, Y.-M.: A double inequality for the gamma and psi functions, Int. J. Mod. Math. 3 (2008), 67-73.Search in Google Scholar

[68] ZHAO, T.-H.-CHU, Y.-M.: A class of logarithmically completely monotonic functions associated with a gamma function, J. Inequal. Appl. 2010 (2010), Article ID 392431, 11 pp.; http://dx.doi.org/10.1155/2010/392431.10.1155/2010/392431Search in Google Scholar

[69] ZHAO, T.-H.-CHU, Y.-M.-JIANG, Y.-P.: Monotonic and logarithmically convex properties of a function involving gamma functions, J. Inequal. Appl. 2009 (2009), Article ID 728612, 13 pp.; http://dx.doi.org/10.1155/2009/728612.10.1155/2009/728612Search in Google Scholar

[70] ZHAO, T.-H.-CHU, Y.-M.-WANG, H.: Logarithmically complete monotonicity properties relating to the gamma function, Abstr. Appl. Anal. 2011 (2011), Article ID 896483, 13 pp.; http://dx.doi.org/10.1155/2011/896483.10.1155/2011/896483Search in Google Scholar

[71] ZHAO, J.-L.-GUO, B.-N.-QI, F.: A refinement of a double inequality for the gamma function, Publ. Math. Debrecen 80 (2012), 333-342; http://dx.doi.org/10.5486/PMD.2012.5010.10.5486/PMD.2012.5010Search in Google Scholar

[72] ZHAO, J.-L.-GUO, B.-N.-QI, F.: Complete monotonicity of two functions involving the tri- and tetra-gamma functions, Period. Math. Hungar. 65 (2012), 147-155; <http://dx.doi.org/10.1007/s10998-012-9562-x>.Search in Google Scholar

[73] WIDDER, D. V.: The Laplace Transform, Princeton University Press, Princeton, NJ, 1946. Search in Google Scholar