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)

 
 

 

Inequalities for the gamma function


Author: Horst Alzer
Journal: Proc. Amer. Math. Soc. 128 (2000), 141-147
MSC (1991): Primary 33B15; Secondary 26D07
DOI: https://doi.org/10.1090/S0002-9939-99-04993-X
Published electronically: June 30, 1999
MathSciNet review: 1622757
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the following two theorems:

(i) Let $M_r(a,b)$ be the $r$th power mean of $a$ and $b$. The inequality

\begin{displaymath}M_r(\Gamma(x),\Gamma(1/x))\ge 1 \end{displaymath}

holds for all $x\in(0,\infty)$ if and only if $r\ge 1/C-\pi^2/(6C^2)$, where $C$ denotes Euler's constant. This refines results established by W. Gautschi (1974) and the author (1997).

(ii) The inequalities

\begin{equation*}x^{\alpha(x-1)-C}<\Gamma(x)<x^{\beta(x-1)-C}\tag{$*$} \end{equation*}

are valid for all $x\in(0,1)$ if and only if $\alpha\le 1-C$ and $\beta\ge (\pi^2/6-C)/2$, while $(*)$ holds for all $x\in (1,\infty)$ if and only if $\alpha\le (\pi^2/6-C)/2$ and $\beta\ge 1$. These bounds for $\Gamma(x)$ improve those given by G. D. Anderson an S.-L. Qiu (1997).


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 94b:00012
  • 2. H. Alzer, A harmonic mean inequality for the gamma function, J. Comput. Appl. Math. 87 (1997), 195-198. CMP 98:06
  • 3. H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), 373-389. MR 97e:33004
  • 4. G. D. Anderson and S.-L. Qiu, A monotoneity property of the gamma function, Proc. Amer. Math. Soc. 125 (1997), 3355-3362. MR 98h:33001
  • 5. P. S. Bullen, D. S. Mitrinovi\'{c}, and P. M. Vasi\'{c}, Means and Their Inequalities, Reidel, Dordrecht, 1988. MR 89d:26003
  • 6. P. J. Davis, Leonhard Euler's integral: A historical profile of the gamma function, Amer. Math. Monthly 66 (1959), 849-869. MR 21:5540
  • 7. W. Gautschi, A harmonic mean inequality for the gamma function SIAM J. Math. Anal. 5 (1974), 278-281. MR 50:2570
  • 8. W. Gautschi, Some mean value inequalities for the gamma function, SIAM J. Math. Anal. 5 (1974), 282-292. MR 50:2571
  • 9. L. Gordon, A stochastic approach to the gamma function, Amer. Math. Monthly 101 (1994), 858-865. MR 95k:33003
  • 10. A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York, 1973. MR 56:1201

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 33B15, 26D07

Retrieve articles in all journals with MSC (1991): 33B15, 26D07


Additional Information

Horst Alzer
Affiliation: Morsbacher Str. 10, 51545 Waldbröl, Germany

DOI: https://doi.org/10.1090/S0002-9939-99-04993-X
Keywords: Gamma function, psi function, power mean, inequalities
Received by editor(s): March 10, 1998
Published electronically: June 30, 1999
Communicated by: Hal L. Smith
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society