Monotonicity of some functions involving the gamma and PSI functions

Mathematics of Computation

Volumn 77, Issue 264, 2008, Pages 2261-2275

  Koumandos, Stamatis ^a  

^a UNIVERSITY OF CYPRUS   (Cyprus)

Author keywords

Completely monotonic functions; Gamma function; psi function

Indexed keywords

EID: 57049150292     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-08-02140-6     Document Type: Article

References (27)

1. M. Abramowitz and I. A. Stegun (Eds). Handbook of Mathematical Functions with formulas, Graphs and Mathematical Tables, Dover. New York, 1965. MR1225604 (94b:00012)
2. H. Alzer, On some inequalities for the gamma and psi functions. Math. Comp. 66, no. 217, (1997) 373-389. MR1388887 (97e:33004)
3. H. Alzer, C. Berg and S. Koumandos, On a conjecture of Clark and Ismail. J. Approx. Theory 134, (2005), no. 1, 102-113. MR2137558 (2006c:33003)
4. H. Alzer, Sharp inequalities for the digamma and polygamma functions. Forum Math. 16 (2004), 181-221. MR2039096 (2005d:33003)
5. H. Alzer and C. Berg, Some classes of completely monotonie functions. II. Ramanujan J. 11, (2006), 225-248. MR2267677 (2007k:33001)
6. H. Alzer and A. Z. Grinshpan, Inequalities for gamma and q-gamma functions. J. Approx. Theory 144 (2007), no. 1, 67-83. MR2287377 (2007m:33004)
7. G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for zero-Balanced hypergeometric functions. Trans. Amer. Math. Soc. 347, no. 5. (1995), 1713-1723. MR1264800 (95m:33002)
8. G. E. Andrews, R. Askey, and R. Roy, Special functions, Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)
9. N. Batir, An interesting double inequality for Eulor's gamma function. JIPAM J. Inequal. Pure Appl. Math. 5 (4) (2004) Article 97. MR2112450 (2005h:33001)
10. N. Batir, Some new inequalities for Gamma and polygamma functions. JIPAM J. Inequal. Pure Appl. Math. 6 (4) (2005) Article 103. MR2178284 (2006k:33001)
11. N. Batir, On some properties of digamma and polygamma functions. J. Math. Anal. Appl. 328 (2007) 452-465. MR2285562
12. J. Bustoz and M. E. H. Ismail, On gamma function inequalities. Math. Comp., 47, no. 176 (1986), 659-667. MR856710 (87m:33002)
13. J. Dubourdieu, Sur un théorème do M. S. Bernstein relatif ála transformation de Laplacc-Stieltjes. Compositio Math. 7 (1939), 96-111. MR0000436 (1:73h)
14. N. Elezović, C. Giordano and J. Pečarić, The best bounds in Gautschi's inequality, Math. Inequal. Appl. 3 (2000), no. 2, 239-252. MR1749300 (2001g:33001)
15. W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38, (1959), 77-81. MR0103289 (21:2067)
16. C. Giordano, A. Laforgia and J. Pečarić, Unified treatment of Gautschi-Kershaw type in-equalities for the gamma function. J. Comp. Appl. Math99, (1998), 167-175. MR1662692 (99k:33004)
17. A. Z. Grinshpan and M. E. H. Ismail, Completely monotonic functions involving the gamma and 9-gamma functions. Proc. Amer. Math. Soc. 134 (2006), no. 4, 1153-1160. MR2196051 (2006i:33003)
18. H. van Haeringen, Completely monotonie and related functions. J. Math. Anal. Appl. 204 (1996), 389-408. MR1421454 (97j:26010)
19. D. Kershaw, Some extensions of W. Gautschi's inequalities for the gamma function. Math. Comp. 41, no. 164, (1983), 607-611. MR717706 (84m:33003)
20. S. Koumandos, Remarks on some completely monotonie functions. J. Math. Anal. Appl., 324 (2006), no. 2, 1458-1461. MR2266574 (2007i:26017)
21. S. Koumandos and S. Ruscheweyh, Positive Gegenbauer polynomial sums and applications to starlike functions. Constr. Approx., 23 (2006), no. 2, 197-210. MR2186305 (2007b:42001)
22. S. Koumandos and S. Ruscheweyh, On a conjecture for trigonometric sums and starlike functions. J. Approx. Theory, 149, no. 1, (2007), 42-58.
23. Feng Qi, Monotonicity and logarithmic convexity for a class of elementary functions involving the exponential function, preprint.
24. Feng Qi, A completely monotonie function involving divided differences of psi and polygamma functions and an application, preprint.
25. S. -L. Qiu and M. Vuorinen, Some properties of the gamma and psi functions, with applications, Math. Comp. 74 (2005), 723-742. MR2114645 (2005i:33002)
26. S. Y. Trimble, Jim Wells, and F.T. Wright, Superadditive functions and a statistical application, SIAM J. Math. Anal., 20, no. 5, (1989), 1255-1259. MR1009357 (91a:26019)
27. D. V. Widder, The Laplace transform, Princeton University Press, Princeton, 1946. MR0005923 (3:232d)