참고문헌
- 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.
- 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.
- H. Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), no. 201, 337-346. https://doi.org/10.2307/2153171
- 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
- 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
- H. Alzer, Inequalities for the gamma and polygamma functions, Abh. Math. Sem. Univ. Hamburg 68 (1998), 363-372. https://doi.org/10.1007/BF02942573
- 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
- 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
- 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
- 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
- H. Alzer and C. Berg, Some classes of completely monotonic functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 445-460.
- G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Special functions of quasi-conformal theory, Exposition. Math. 7 (1989), no. 2, 97-136.
- 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.
- 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
- S. Bochner, Harmonic Analysis and the Theory of Probability, University of California Press, Berkeley and Los Angeles, 1955.
- L. Bougoffa, Some inequalities involving the gamma function, JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 5, Article 179, 3 pp.
- 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
- 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
- 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
- M. J. Dubourdieu, Sur un theoreme de M. S. Bernstein relatif a la transformation de Laplace-Stieltjes, Compositio Math. 7 (1939), 96-111.
- 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
- W. Gautschi, A harmonic mean inequality for the gamma function, SIAM J. Math. Anal. 5 (1974), 278-281. https://doi.org/10.1137/0505030
- W. Gautschi, Some mean value inequalities for the gamma function, SIAM J. Math. Anal. 5 (1974), 282-292. https://doi.org/10.1137/0505031
- 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
- 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
- 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
- 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.
- Group of compilation, Handbook of Mathematics, Peoples' Education Press, Beijing, China, 1979 (Chinese).
- G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge, at the University Press, 1952.
- R. A. Horn, On innitely divisible matrices, kernels, and functions, Z. Wahrsch. Verw. Gebiete 8 (1967), 219-230. https://doi.org/10.1007/BF00531524
- 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.
- 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).
- 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.
- V. Krasniqi and F. Merovci, Generalization of some inequalities for the special functions, J. Inequal. Spec. Funct. 3 (2012), no. 3, 34-40.
- 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.
- W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer, Berlin, 1966.
- 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.
- 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.
- 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.
- 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
- 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.
- 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
- 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
- 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.
- 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
- 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
- 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
- J. Sandor, On convex functions involving Euler's Gamma function, Math. Mag. 8 (2000), 514-515.
- 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.
- A. Sh. Shabani, Some inequalities for the gamma function, J. Inequal. Pure Appl. Math. 8 (2007), no. 2, Article 49, 4 pp.
- A. Sh. Shabani, Generalization of some inequalities for the gamma function, Math. Commun. 13 (2008), no. 2, 271-275.
- 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
- W. R. Wade, An Introduction to Analysis, 4th Edi., Pearson Education International, Prentice Hall, 2010.
- D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1941.