×

A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw’s double inequality. (English) Zbl 1145.33001

Two theorems are presented establishing necessary and sufficient conditions that a class of functions involving the psi function and the ratio of particular gamma functions are logarithmically completely monotonic. The proofs of these theorems are given by first proving an important three part proposition results on \(f\)-means and their applications to the digamma function given by N. Elezovic and J. Pecaric [Math. Inequal. Appl. 3, No. 2, 189–196 (2000; Zbl 0952.26010)].

MSC:

33B15 Gamma, beta and polygamma functions
44A10 Laplace transform
65R10 Numerical methods for integral transforms
26A48 Monotonic functions, generalizations
26A51 Convexity of real functions in one variable, generalizations
26D20 Other analytical inequalities

Citations:

Zbl 0952.26010
PDFBibTeX XMLCite
Full Text: DOI

References:

[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, Washington, 1970 (9th printing).; 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, Washington, 1970 (9th printing).
[2] Alzer, H., Mean-value inequalities for the polygamma functions, Aequationes Math., 61, 151-161 (2001) · Zbl 0968.33003
[3] Atanassov, R. D.; Tsoukrovski, U. V., Some properties of a class of logarithmically completely monotonic functions, C.R. Acad. Bulgare Sci., 41, 2, 21-23 (1988) · Zbl 0658.26010
[4] Batir, N., Some gamma function inequalities, RGMIA Res. Rep. Coll., 9, 3 (2006), Art. 5. Available online at \(\langle\) http://rgmia.vu.edu.au/v9n3.html \(\rangle \)
[5] Berg, C., Integral representation of some functions related to the gamma function, Mediterr. J. Math., 1, 4, 433-439 (2004) · Zbl 1162.33300
[6] Bustoz, J.; Ismail, M. E.H., On gamma function inequalities, Math. Comput., 47, 659-667 (1986) · Zbl 0607.33002
[7] Cargo, G. T., Comparable means and generalized convexity, J. Math. Anal. Appl., 12, 387-392 (1965) · Zbl 0144.30504
[8] Chen, Ch.-P., Monotonicity and convexity for the gamma function, J. Inequal. Pure Appl. Math., 6, 4 (2005), Art. 100. Available online at \(\langle\) http://jipam.vu.edu.au/article.php?\(sid=574 \rangle \) · Zbl 1081.33004
[9] Chen, Ch.-P.; Qi, F., Logarithmically complete monotonicity properties for the gamma functions, Austral. J. Math. Anal. Appl., 2, 2 (2005), Art. 8. Available online at \(\langle\) http://ajmaa.org/cgi-bin/paper.pl?string=v2n2/V2I2P8.tex \(\rangle \) · Zbl 1090.33002
[10] Chen, Ch.-P.; Qi, F., Logarithmically completely monotonic ratios of mean values and an application, Glob. J. Math. Math. Sci., 1, 1, 71-76 (2005), RGMIA Res. Rep. Coll. 8 (1) (2005), Art. 18, 147-152. Available online at \(\langle\) http://rgmia.vu.edu.au/v8n1.html \(\rangle \)
[11] Chen, Ch.-P.; Qi, F., Logarithmically completely monotonic functions relating to the gamma function, J. Math. Anal. Appl., 321, 1, 405-411 (2006) · Zbl 1099.33002
[12] Elezović, N.; Giordano, C.; Pečarić, J., The best bounds in Gautschi’s inequality, Math. Inequal. Appl., 3, 239-252 (2000) · Zbl 0947.33001
[13] Elezović, N.; Pečarić, J., Differential and integral \(f\)-means and applications to digamma function, Math. Inequal. Appl., 3, 2, 189-196 (2000) · Zbl 0952.26010
[14] Erber, T., The gamma function inequalities of Gurland and Gautschi, Scand. Actuar. J., 1960, 27-28 (1961) · Zbl 0104.29204
[15] Gautschi, W., Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys., 38, 77-81 (1959/1960) · Zbl 0094.04104
[16] Grinshpan, A. Z.; Ismail, M. E.H., Completely monotonic functions involving the gamma and \(q\)-gamma functions, Proc. Amer. Math. Soc., 134, 1153-1160 (2006) · Zbl 1085.33001
[17] Guo, S., Monotonicity and concavity properties of some functions involving the gamma function with applications, J. Inequal. Pure Appl. Math., 7, 2 (2006), Art. 45. Available online at \(\langle\) http://jipam.vu.edu.au/article.php?\(sid=662 \rangle \) · Zbl 1133.33002
[18] Guo, B.-N.; Qi, F., Two classes of completely monotonic functions involving gamma and polygamma functions, RGMIA Res. Rep. Coll., 8, 3 (2005), Art. 16. Available online at \(\langle\) http://rgmia.vu.edu.au/v8n3.html \(\rangle \)
[19] Horn, R. A., On infinitely divisible matrices, kernels and functions, Z. Wahrscheinlichkeitstheorie Verw. Geb., 8, 219-230 (1967) · Zbl 0314.60017
[20] Kečlić, J. D.; Vasić, P. M., Some inequalities for the gamma function, Publ. Inst. Math. (Beograd) (N.S.), 11, 107-114 (1971) · Zbl 0222.33001
[21] Kershaw, D., Some extensions of W. Gautschi’s inequalities for the gamma function, Math. Comput., 41, 607-611 (1983) · Zbl 0536.33002
[22] Laforgia, A., Further inequalities for the gamma function, Math. Comput., 42, 166, 597-600 (1984) · Zbl 0536.33003
[23] Li, A.-J.; Zhao, W.-Zh.; Chen, Ch.-P., Logarithmically complete monotonicity and Schur-convexity for some ratios of gamma functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 17, 88-92 (2006) · Zbl 1150.33301
[24] Mitrinović, D. S.; Pečarić, J. E.; Fink, A. M., Classical and New Inequalities in Analysis (1993), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 0771.26009
[25] Qi, F., Generalized weighted mean values with two parameters, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454, 1978, 2723-2732 (1998) · Zbl 0935.26014
[26] Qi, F., Generalized abstracted mean values, J. Inequal. Pure Appl. Math., 1, 1 (2000), Art. 4. Available online at \(\langle\) http://jipam.vu.edu.au/article.php?\(sid=97 \rangle \), RGMIA Res. Rep. Coll. 2 (5) (1999), Art. 4, 633-642. Available online at \(\langle\) http://rgmia.vu.edu.au/v2n5.html \(\rangle \) · Zbl 0989.26020
[27] Qi, F., Monotonicity results and inequalities for the gamma and incomplete gamma functions, Math. Inequal. Appl., 5, 1, 61-67 (2002), RGMIA Res. Rep. Coll. 2 (7) (1999), Art. 7, 1027-1034. Available online at \(\langle\) http://rgmia.vu.edu.au/v2n7.html \(\rangle \) · Zbl 1006.33001
[28] Qi, F., The extended mean values: definition, properties, monotonicities, comparison, convexities, generalizations, and applications, Cubo Mat. Ed., 5, 3, 63-90 (2003), RGMIA Res. Rep. Coll. 5 (1) (2002), Art. 5, 57-80. Available online at \(\langle\) http://rgmia.vu.edu.au/v5n1.html \(\rangle \) · Zbl 1162.26317
[29] Qi, F., Certain logarithmically \(N\)-alternating monotonic functions involving gamma and \(q\)-gamma functions, RGMIA Res. Rep. Coll., 8, 3 (2005), Art. 5. Available online at \(\langle\) http://rgmia.vu.edu.au/v8n3.html \(\rangle \)
[30] 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), doi: \( \langle;\) http://dx.doi.org/10.1016/j.cam.\(2006.09.005 \rangle;\). RGMIA Res. Rep. Coll. 9 (2) (2006), Art. 16. Available online at \(\langle;\) http://rgmia.vu.edu.au/v9n2.html \(\rangle;\).; 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), doi: \( \langle;\) http://dx.doi.org/10.1016/j.cam.\(2006.09.005 \rangle;\). RGMIA Res. Rep. Coll. 9 (2) (2006), Art. 16. Available online at \(\langle;\) http://rgmia.vu.edu.au/v9n2.html \(\rangle;\).
[31] Qi, F., A completely monotonic function involving divided difference of psi function and an equivalent inequality involving sum, RGMIA Res. Rep. Coll., 9, 4 (2006), Art. 5. Available online at \(\langle\) http://rgmia.vu.edu.au/v9n4.html \(\rangle \)
[32] Qi, F., A completely monotonic function involving divided differences of psi and polygamma functions and an application, RGMIA Res. Rep. Coll., 9, 4 (2006), Art. 8. Available online at \(\langle\) http://rgmia.vu.edu.au/v9n4.html \(\rangle \)
[33] Qi, F., The best bounds in Kershaw’s inequality and two completely monotonic functions, RGMIA Res. Rep. Coll., 9, 4 (2006), Art. 2. Available online at \(\langle\) http://rgmia.vu.edu.au/v9n4.html \(\rangle \)
[34] F. Qi, A class of logarithmically completely monotonic functions and application to the best bounds in the second Gautschi-Kershaw’s inequality, Comput. Math. Appl. (2007), in press. RGMIA Res. Rep. Coll. 9 (4) (2006), Art. 11. Available online at \(\langle;\) http://rgmia.vu.edu.au/v9n4.html \(\rangle;\).; F. Qi, A class of logarithmically completely monotonic functions and application to the best bounds in the second Gautschi-Kershaw’s inequality, Comput. Math. Appl. (2007), in press. RGMIA Res. Rep. Coll. 9 (4) (2006), Art. 11. Available online at \(\langle;\) http://rgmia.vu.edu.au/v9n4.html \(\rangle;\).
[35] F. Qi, A property of logarithmically absolutely monotonic functions and logarithmically complete monotonicities of \((1 + \operatorname{Α;} / x )^{x + \operatorname{Β;}} \), Integral Transform Spec. Funct. (2007), accepted for publication.; F. Qi, A property of logarithmically absolutely monotonic functions and logarithmically complete monotonicities of \((1 + \operatorname{Α;} / x )^{x + \operatorname{Β;}} \), Integral Transform Spec. Funct. (2007), accepted for publication.
[36] F. Qi, Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct. 18 (2007), accepted for publication. RGMIA Res. Rep. Coll. 9 (2006), Suppl., Art. 6. Available online at \(\langle;\) http://rgmia.vu.edu.au/v9(E).html \(\rangle;\).; F. Qi, Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct. 18 (2007), accepted for publication. RGMIA Res. Rep. Coll. 9 (2006), Suppl., Art. 6. Available online at \(\langle;\) http://rgmia.vu.edu.au/v9(E).html \(\rangle;\).
[37] F. Qi, Wendel-Gautschi-Kershaw’s inequalities and sufficient and necessary conditions that a class of functions involving ratio of gamma functions are logarithmically completely monotonic, Math. Comp. (2007), accepted for publication. RGMIA Res. Rep. Coll. 10 (1) (2007), Available online at \(\langle;\) http://rgmia.vu.edu.au/v10n1.html \(\rangle;\).; F. Qi, Wendel-Gautschi-Kershaw’s inequalities and sufficient and necessary conditions that a class of functions involving ratio of gamma functions are logarithmically completely monotonic, Math. Comp. (2007), accepted for publication. RGMIA Res. Rep. Coll. 10 (1) (2007), Available online at \(\langle;\) http://rgmia.vu.edu.au/v10n1.html \(\rangle;\).
[38] Qi, F.; Cao, J.; Niu, D.-W., Four logarithmically completely monotonic functions involving gamma function and originating from problems of traffic flow, RGMIA Res. Rep. Coll., 9, 3 (2006), Art 9. Available online at \(\langle\) http://rgmia.vu.edu.au/v9n3.html \(\rangle \)
[39] Qi, F.; Chen, Ch.-P., A complete monotonicity property of the gamma function, J. Math. Anal. Appl., 296, 2, 603-607 (2004) · Zbl 1046.33001
[40] F. Qi, W.-S. Cheung, Logarithmically completely monotonic functions concerning gamma and digamma functions, Integral Transform Spec. Funct. 18 (2007), in press.; F. Qi, W.-S. Cheung, Logarithmically completely monotonic functions concerning gamma and digamma functions, Integral Transform Spec. Funct. 18 (2007), in press. · Zbl 1120.33003
[41] Qi, F.; Guo, B.-N., Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll., 7, 1, 63-72 (2004), Art. 8, Available online at \(\langle\) http://rgmia.vu.edu.au/v7n1.html \(\rangle \)
[42] Qi, F.; Guo, B.-N.; Chen, Ch.-P., Some completely monotonic functions involving the gamma and polygamma functions, RGMIA Res. Rep. Coll., 7, 1, 31-36 (2004), Art. 5, Available online at \(\langle\) http://rgmia.vu.edu.au/v7n1.html \(\rangle \)
[43] Qi, F.; Guo, B.-N.; Chen, Ch.-P., Some completely monotonic functions involving the gamma and polygamma functions, J. Austral. Math. Soc., 80, 81-88 (2006) · Zbl 1094.33002
[44] Qi, F.; Guo, B.-N.; Chen, Ch.-P., The best bounds in Gautschi-Kershaw inequalities, Math. Inequal. Appl., 9, 3, 427-436 (2006), RGMIA Res. Rep. Coll. 8 (2) (2005), Art. 17. Available online at \(\langle\) http://rgmia.vu.edu.au/v8n2.html \(\rangle \) · Zbl 1101.33001
[45] Qi, F.; Li, W.; Guo, B.-N., Generalizations of a theorem of I. Schur, RGMIA Res. Rep. Coll., 9, 3 (2006), Art. 15. Available online at \(\langle\) http://rgmia.vu.edu.au/v9n3.html \(\rangle \). Bùděngshì Yānjiū Tōngxùn (Commun. Stud. Inequal.) 13 (4) (2006) 355-364
[46] Qi, F.; Niu, D.-W.; Cao, J., Logarithmically completely monotonic functions involving gamma and polygamma functions, J. Math. Anal. Approx. Theory, 1, 1, 66-74 (2006), RGMIA Res. Rep. Coll. 9 (1) (2006), Art. 15. Available online at \(\langle\) http://rgmia.vu.edu.au/v9n1.html \(\rangle \) · Zbl 1113.33005
[47] Qi, F.; Wei, Z.-L.; Yang, Q., Generalizations and refinements of Hermite-Hadamard’s inequality, Rocky Mountain J. Math., 35, 1, 235-251 (2005), RGMIA Res. Rep. Coll. 5 (2) (2002), Art. 10, 337-349. Available online at \(\langle\) http://rgmia.vu.edu.au/v5n2.html \(\rangle \) · Zbl 1096.26014
[48] Qi, F.; Xu, S.-L., The function \((b^x - a^x) / x\): inequalities and properties, Proc. Amer. Math. Soc., 126, 11, 3355-3359 (1998) · Zbl 0904.26006
[49] Qi, F.; Yang, M.-L., Comparisons of two integral inequalities with Hermite-Hadamard-Jensen’s integral inequality, Octogon Math. Mag., 14, 1, 53-58 (2006), RGMIA Res. Rep. Coll. 8 (3) (2005), Art. 18. Available online at \(\langle\) http://rgmia.vu.edu.au/v8n3.html \(\rangle \)
[50] Qi, F.; Yang, Q.; Li, W., Two logarithmically completely monotonic functions connected with gamma function, Integral Transform Spec. Funct., 17, 7, 539-542 (2006), RGMIA Res. Rep. Coll. 8 (3) (2005), Art. 13. Available online at \(\langle\) http://rgmia.vu.edu.au/v8n3.html \(\rangle \) · Zbl 1093.33001
[51] Qi, F.; Zhang, Sh.-Q., Note on monotonicity of generalized weighted mean values, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455, 1989, 3259-3260 (1999) · Zbl 0942.26030
[52] H. van Haeringen, Completely monotonic and related functions, Report 93-108, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands, 1993.; H. van Haeringen, Completely monotonic and related functions, Report 93-108, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands, 1993. · Zbl 0889.26008
[53] Wendel, J. G., Note on the gamma function, Amer. Math. Monthly, 55, 9, 563-564 (1948)
[54] Widder, D. V., The Laplace Transform (1941/1946), Princeton University Press: Princeton University Press Princeton · Zbl 0060.24801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.