1 |
W. R. Wade, An Introduction to Analysis, 4th Edi., Pearson Education International, Prentice Hall, 2010.
|
2 |
W. Gautschi, A harmonic mean inequality for the gamma function, SIAM J. Math. Anal. 5 (1974), 278-281. https://doi.org/10.1137/0505030
DOI
|
3 |
W. Gautschi, Some mean value inequalities for the gamma function, SIAM J. Math. Anal. 5 (1974), 282-292. https://doi.org/10.1137/0505031
DOI
|
4 |
C. Giordano and A. Laforgia, Inequalities and monotonicity properties for the gamma function, J. Comput. Appl. Math. 133 (2001), no. 1-2, 387-396. https://doi.org/10.1016/S0377-0427(00)00659-2
DOI
|
5 |
P. J. Grabner, R. F. Tichy, and U. T. Zimmermann, Inequalities for the gamma function with applications to permanents, Discrete Math. 154 (1996), no. 1-3, 53-62. https: //doi.org/10.1016/0012-365X(94)00340-O
DOI
|
6 |
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
DOI
|
7 |
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.
|
8 |
Group of compilation, Handbook of Mathematics, Peoples' Education Press, Beijing, China, 1979 (Chinese).
|
9 |
G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge, at the University Press, 1952.
|
10 |
R. A. Horn, On innitely divisible matrices, kernels, and functions, Z. Wahrsch. Verw. Gebiete 8 (1967), 219-230. https://doi.org/10.1007/BF00531524
DOI
|
11 |
H.-H. Kairies, An inequality for Krull solutions of a certain dierence equation, in General inequalities, 3 (Oberwolfach, 1981), 277-280, Internat. Schriftenreihe Numer. Math., 64, Birkhauser, Basel, 1983.
|
12 |
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
DOI
|
13 |
D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1941.
|
14 |
M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC, 1964.
|
15 |
C. Alsina and M. S. Tomas, A geometrical proof of a new inequality for the gamma function, J. Ineq. Pure Appl. Math. 6 (2005), no. 2, Article. 48. Available online at https://www.emis.de/journals/JIPAM/article517.html.
|
16 |
H. Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), no. 201, 337-346. https://doi.org/10.2307/2153171
DOI
|
17 |
H. Alzer, A harmonic mean inequality for the gamma function, J. Comput. Appl. Math. 87 (1997), no. 2, 195-198. https://doi.org/10.1016/S0377-0427(96)00181-1
DOI
|
18 |
H. Alzer, Inequalities for the gamma and polygamma functions, Abh. Math. Sem. Univ. Hamburg 68 (1998), 363-372. https://doi.org/10.1007/BF02942573
DOI
|
19 |
H. Alzer, Inequalities for the gamma function, Proc. Amer. Math. Soc. 128 (2000), no. 1, 141-147. https://doi.org/10.1090/S0002-9939-99-04993-X
DOI
|
20 |
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
DOI
|
21 |
H. Alzer, On Gautschi's harmonic mean inequality for the gamma function, J. Comput. Appl. Math. 157 (2003), no. 1, 243-249. https://doi.org/10.1016/S0377-0427(03)00456-4
DOI
|
22 |
H. Alzer, Inequalities for the volume of the unit ball in Rn. II, Mediterr. J. Math. 5 (2008), no. 4, 395-413. https://doi.org/10.1007/s00009-008-0158-x
DOI
|
23 |
A. McD. Mercer, Some new inequalities for the gamma, beta and zeta functions, J. Inequal. Pure Appl. Math. 7 (2006), no. 1, Article 29, 6 pp.
|
24 |
D. Kershaw and A. Laforgia, Monotonicity results for the gamma function, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 119 (1985), no. 3-4, 127-133 (1986).
|
25 |
T. Kim and C. Adiga, On the q-analogue of gamma functions and related inequalities, JIPAM. J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Article 118, 4 pp.
|
26 |
H. Alzer and C. Berg, Some classes of completely monotonic functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 445-460.
|
27 |
V. Krasniqi and F. Merovci, Generalization of some inequalities for the special functions, J. Inequal. Spec. Funct. 3 (2012), no. 3, 34-40.
|
28 |
A. Laforgia and S. Sismondi, A geometric mean inequality for the gamma function, Boll. Un. Mat. Ital. A (7) 3 (1989), no. 3, 339-342.
|
29 |
W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer, Berlin, 1966.
|
30 |
T. Mansour, Some inequalities for the q-gamma function, J. Inequal. Pure Appl. Math. 9 (2008), no. 1, Article 18. Available online at http://www.emis.de/journals/JIPAM/article954.html.
|
31 |
E. Neuman, Inequalities involving a logarithmically convex function and their applications to special functions, J. Inequal. Pure Appl. Math. 7 (2006), no. 1, Article 16, 4 pp.
|
32 |
E. Neuman, Some inequalities for the gamma function, Appl. Math. Comput. 218 (2011), no. 8, 4349-4352. https://doi.org/10.1016/j.amc.2011.10.010
DOI
|
33 |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC, 2010.
|
34 |
F. Qi and R. P. Agarwal, On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequal. Appl. 2019 (2019), Paper No. 36, 42 pp. https://doi.org/10.1186/s13660-019-1976-z
DOI
|
35 |
S. Bochner, Harmonic Analysis and the Theory of Probability, University of California Press, Berkeley and Los Angeles, 1955.
|
36 |
G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Special functions of quasi-conformal theory, Exposition. Math. 7 (1989), no. 2, 97-136.
|
37 |
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.
|
38 |
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
DOI
|
39 |
L. Bougoffa, Some inequalities involving the gamma function, JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 5, Article 179, 3 pp.
|
40 |
C.-P. Chen, Complete monotonicity and logarithmically complete monotonicity proper- ties for the gamma and psi functions, J. Math. Anal. Appl. 336 (2007), no. 2, 812-822. https://doi.org/10.1016/j.jmaa.2007.03.028
DOI
|
41 |
C.-P. Chen, F. Qi, and H. M. Srivastava, Some properties of functions related to the gamma and psi functions, Integral Transforms Spec. Funct. 21 (2010), no. 1-2, 153-164. https://doi.org/10.1080/10652460903064216
DOI
|
42 |
C.-P. Chen and H. M. Srivastava, Some inequalities and monotonicity properties associated with the gamma and psi functions and the Barnes G-function, Integral Transforms Spec. Funct. 22 (2011), no. 1, 1-15. https://doi.org/10.1080/10652469.2010.483899
DOI
|
43 |
M. J. Dubourdieu, Sur un theoreme de M. S. Bernstein relatif a la transformation de Laplace-Stieltjes, Compositio Math. 7 (1939), 96-111.
|
44 |
P. Gao, Some monotonicity properties of gamma and q-gamma functions, ISRN Math. Anal. 2011 (2011), Art. ID 375715, 15 pp. https://doi.org/10.5402/2011/375715
DOI
|
45 |
F. Qi, B.-N. Guo, S. Guo, and Sh.-X. Chen, A function involving gamma function and having logarithmically absolute convexity, Integral Transforms Spec. Funct. 18 (2007), no. 11-12, 837-843. https://doi.org/10.1080/10652460701528875
DOI
|
46 |
F. Qi and C.-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
DOI
|
47 |
F. Qi and B.-N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Article 8, 63-72; Available online at http://rgmia.org/v7n1.php.
|
48 |
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), no. 1, 81-88. https://doi.org/10.1017/S1446788700011393
DOI
|
49 |
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
DOI
|
50 |
J. Sandor, On convex functions involving Euler's Gamma function, Math. Mag. 8 (2000), 514-515.
|
51 |
J. Sandor, A note on certain inequalities for the gamma function, J. Ineq. Pure Appl. Math. 6 (2005), no. 3, Article. 61. Available online at https://www.emis.de/journals/JIPAM/article534.html.
|
52 |
A. Sh. Shabani, Some inequalities for the gamma function, J. Inequal. Pure Appl. Math. 8 (2007), no. 2, Article 49, 4 pp.
|
53 |
A. Sh. Shabani, Generalization of some inequalities for the gamma function, Math. Commun. 13 (2008), no. 2, 271-275.
|
54 |
H. M. Srivastava and J. Choi, Zeta and q-Zeta functions and Associated Series and Integrals, Elsevier, Inc., Amsterdam, 2012. https://doi.org/10.1016/B978-0-12-385218-2.00001-3
|