References
- H. Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), no. 201, 337–346. https://doi.org/10.1090/S0025-5718-1993-1149288-7
- 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 volume of the unit ball in \
$R^n$ , J. Math. Anal. Appl. 252 (2000), no. 1, 353–363. https://doi.org/10.1006/jmaa.2000.7065 - H. Alzer, On an inequality of H. Minc and L. Sathre, J. Math. Anal. Appl. 179 (1993), no. 2, 396–402. https://doi.org/10.1006/jmaa.1993.1358
- G. D. Anderson and S. L. Qiu, A monotoneity property of the gamma function, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3355–3362. https://doi.org/10.1090/S0002-9939-97-04152-X
- A. Elbert and A. Laforgia, On some properties of the gamma function, Proc. Amer. Math. Soc. 128 (2000), no. 9, 2667–2673. https://doi.org/10.1090/S0002-9939-00-05520-9
- G. M. Fichtenholz, Differential- und Integralrechnung. II, VEB Deutscher Verlag der Wissenschaften, Berlin, 1986.
- C. E. Finol and M. Wojtowicz, Multiplicative properties of real functions with applications to classical functions, Aequationes Math. 59 (2000), no. 1-2, 134–149. https://doi.org/10.1007/PL00000120
- D. Kershaw, Some extensions of W. Gautschi’s inequalities for the gamma function, Math. Comp. 41 (1983), no. 164, 607–611.
-
J. Matkowski,
$L^p-like$ paranorms, Selected topics in functional equations and iteration theory (Graz, 1991), 103–138, Grazer Math. Ber., 316, Karl-Franzens-Univ. Graz, Graz, 1992. - M. Merkle, Logarithmic convexity and inequalities for the gamma function, J. Math. Anal. Appl. 203 (1996), no. 2, 369–380. https://doi.org/10.1006/jmaa.1996.0385
-
H. Minc and L. Sathre, Some inequalities involving
$(r!)^{1/r}$ , Proc. Edinburgh Math. Soc. (2) 14 (1964/1965), 41–46. https://doi.org/10.1017/S0013091500011214 - P. Montel, Sur les functions convexes et les fonctions sousharmoniques, J. Math. Pures Appl. (9) 7 (1928), 29–60.
- C. P. Niculescu, Convexity according to the geometric mean, Math. Inequal. Appl. 3 (2000), no. 2, 155–167.
- C. P. Niculescu, Convexity according to means, Math. Inequal. Appl. 6 (2003), no. 4, 571–579.
- B. Palumbo, A generalization of some inequalities for the gamma function, J. Comput. Appl. Math. 88 (1998), no. 2, 255–268. https://doi.org/10.1016/S0377-0427(97)00187-8
- J. Pecaric, G. Allasia, and C. Giordano, Convexity and the gamma function, Indian J. Math. 41 (1999), no. 1, 79–93.
- 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 C. P. Chen, Monotonicity and convexity results for functions involving the gamma function, Int. J. Appl. Math. Sci. 1 (2004), 27–36.
- F. Qi, B. N. Guo, and C. P. Chen, The best bounds in Gautschi-Kershaw inequalities, Math. Inequal. Appl. 9 (2006), no. 3, 427–436.
- E. T. Whittaker and G. N.Watson, A Course of Modern Analysis, Cambridge University Press, New York, 1962.
Cited by
- Necessary and sufficient conditions for a class of functions and their reciprocals to be logarithmically completely monotonic vol.2011, pp.1, 2011, https://doi.org/10.1186/1029-242X-2011-36
- Logarithmically Complete Monotonicity Properties Relating to the Gamma Function vol.2011, 2011, https://doi.org/10.1155/2011/896483
- A Class of Logarithmically Completely Monotonic Functions Associated with a Gamma Function vol.2010, pp.1, 2010, https://doi.org/10.1155/2010/392431