Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Sub- and superadditive properties of Euler's gamma function


Author: Horst Alzer
Journal: Proc. Amer. Math. Soc. 135 (2007), 3641-3648
MSC (2000): Primary 33B15, 39B62; Secondary 26D15
DOI: https://doi.org/10.1090/S0002-9939-07-09057-0
Published electronically: August 6, 2007
MathSciNet review: 2336580
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \alpha>0$ and $ 0<c \neq 1$ be real numbers. The inequality

$\displaystyle \Bigl(\frac{\Gamma(x+y+c)}{\Gamma(x+y)}\Bigr)^{1/\alpha}< \Bigl(\... ...mma(x)}\Bigr)^{1/\alpha}+ \Bigl(\frac{\Gamma(y+c)}{\Gamma(y)}\Bigr)^{1/\alpha} $

holds for all positive real numbers $ x, y$ if and only if $ \alpha\geq \max(1,c)$. The reverse inequality is valid for all $ x,y>0$ if and only if $ \alpha\leq \min(1,c)$.


References [Enhancements On Off] (What's this?)

  • [1] M. Abramowitz and I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1965. MR 1225604 (94b:00012)
  • [2] H. Alzer, Inequalities for the Hurwitz zeta function, Proc. Royal Soc. Edinburgh 130A (2000), 1227-1236. MR 1809101 (2001k:11167)
  • [3] H. Alzer and S. Koumandos, Sub- and superadditive properties of Fejér's sine polynomial, Bull. London Math. Soc. 38 (2006), 261-268. MR 2214478 (2006m:42001)
  • [4] H. Alzer and S. Ruscheweyh, A subadditive property of the gamma function, J. Math. Anal. Appl. 285 (2003), 564-577. MR 2005141 (2004i:33002)
  • [5] E.F. Beckenbach, Superadditivity inequalities, Pacific J. Math. 14 (1964), 421-438. MR 0163996 (29:1295)
  • [6] E. Berz, Sublinear functions on $ R$, Aequat. Math. 12 (1975), 200-206. MR 0387862 (52:8700)
  • [7] A. Bruckner, Minimal superadditive extensions of superadditive functions, Pacific J. Math. 10 (1960), 1155-1162. MR 0122943 (23:A275)
  • [8] A.M. Bruckner, Tests for the superadditivity of functions, Proc. Amer. Math. Soc. 13 (1962), 126-130. MR 0133411 (24:A3245)
  • [9] A.M. Bruckner, Some relationships between locally superadditive functions and convex functions, Proc. Amer. Math. Soc. 15 (1964), 61-65. MR 0156924 (28:167)
  • [10] A.M. Bruckner and E. Ostrow, Some function classes related to the class of convex functions, Pacific J. Math. 12 (1962), 1203-1215. MR 0148822 (26:6326)
  • [11] G. Buskes, The Hahn-Banach theorem-surveyed, Dissert. Math. 327 (1993), 49 pp. MR 1240598 (94h:46007)
  • [12] R. Cooper, The converses of the Cauchy-Hölder inequality and the solutions of the inequality $ g(x+y)\leq g(x)+g(y)$, Proc. London Math. Soc. 26 (2) (1927), 415-432.
  • [13] Z. Gajda and Z. Kominek, On separation theorems for subadditive and superadditive functionals, Studia Math. 100 (1991), 25-38. MR 1130135 (93h:39003)
  • [14] R. Garunkštis, On some inequalities concerning $ \pi(x)$, Exper. Math. 11 (2002), 297-301. MR 1959270 (2003k:11143)
  • [15] W. Gautschi, The incomplete gamma function since Tricomi, in: Tricomi's Ideas and Contemporary Applied Mathematics, Atti Convegni Lincei, 147, Accad. Naz. Lincei, Rome, 1998, 203-237. MR 1737497 (2001g:33003)
  • [16] E. Hille and R.S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc., Coll. Publ. 31, Providence R.I., 1957. MR 0089373 (19:664d)
  • [17] R.F. Jolly, Concerning periodic subadditive functions, Pacific J. Math. 15 (1965), 159-169. MR 0176005 (31:281)
  • [18] E. Kamke, Über die eindeutige Bestimmtheit der Integrale von Differentialgleichungen, II. Sitzungsber. Akad. Wiss. Heidelberg, Math.-nat. Kl., 1931, no. 17, 15 pp.
  • [19] R.G. Laatsch, Extensions of subadditive functions, Pacific J. Math. 14 (1964), 209-215. MR 0162897 (29:201)
  • [20] J. Matkowski, Subadditive functions and a relaxation of the homogeneity condition of seminorms, Proc. Amer. Math. Soc. 117 (1993), 991-1001. MR 1113646 (93e:26002)
  • [21] J. Matkowski and T. Swiatkowski, On subadditive functions, Proc. Amer. Math. Soc. 119 (1993), 187-197. MR 1176072 (93k:26002)
  • [22] H. Minkowski, Geometrie der Zahlen, Teubner, Berlin-Leipzig, 1910.
  • [23] D.S. Mitrinovic, Analytic Inequalities, Springer, New York, 1970. MR 0274686 (43:448)
  • [24] L.P. Østerdal, Subadditive functions and their (pseudo-)inverses, J. Math. Anal. Appl. 317 (2006), 724-731. MR 2209592 (2006k:26010)
  • [25] L. Panaitopol, Inequalities concerning the function $ \pi(x)$: applications, Acta Arith. 94 (2000), 373-381. MR 1779949 (2001g:11144)
  • [26] R.A. Rosenbaum, Sub-additive functions, Duke Math. J. 17 (1950), 227-247. MR 0036796 (12:164a)
  • [27] R.M. Tardiff and H.W. Austin, Convexity and a generalized Minkowski inequality, Houston J. Math. 15 (1989), 121-128. MR 1071269 (91h:54012)
  • [28] S.Y. Trimble, J. Wells, and F.T. Wright, Superadditive functions and a statistical application, SIAM J. Math. Anal. 20 (1989), 1255-1259. MR 1009357 (91a:26019)
  • [29] J.S.W. Wong, A note on subadditive functions, Proc. Amer. Math. Soc. 44 (1974), 106. MR 0327985 (48:6327)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 33B15, 39B62, 26D15

Retrieve articles in all journals with MSC (2000): 33B15, 39B62, 26D15


Additional Information

Horst Alzer
Affiliation: Morsbacher Str. 10, D-51545 Waldbröl, Germany
Email: alzerhorst@freenet.de

DOI: https://doi.org/10.1090/S0002-9939-07-09057-0
Keywords: Gamma and psi functions, sub- and superadditive, convex, inequalities.
Received by editor(s): September 5, 2006
Published electronically: August 6, 2007
Communicated by: Andreas Seeger
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society