Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On some inequalities for the incomplete
gamma function

Author: Horst Alzer
Journal: Math. Comp. 66 (1997), 771-778
MSC (1991): Primary 33B20; Secondary 26D07, 26D15
MathSciNet review: 1397438
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $p\ne 1$ be a positive real number. We determine all real numbers $\alpha = \alpha (p)$ and $\beta =\beta (p)$ such that the inequalities

\begin{displaymath}[1-e^{-\beta x^p}]^{1/p}< \frac 1{\Gamma (1+1/p)} \int ^x_0 e^{-t^p} \,dt <[1-e^{-\alpha x^p}]^{1/p}\end{displaymath}

are valid for all $x>0$. And, we determine all real numbers $a$ and $b$ such that

\begin{displaymath}-\log (1-e^{-ax})\le \int ^\infty _x \frac {e^{-t}}t\,dt\le -\log (1-e^{-bx})\end{displaymath}

hold for all $x>0$.

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

  • 1. J. T. Chu, On bounds for the normal integral, Biometrika 42 (1955), 263-265. MR 16:838f
  • 2. G. M. Fichtenholz, Differential- und Integralrechnung, II, Dt. Verlag Wissensch., Berlin, 1979. MR 80f:26001
  • 3. W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys. 38 (1959), 77-81. MR 21:2067
  • 4. D. S. Mitrinovi\'{c}, Analytic inequalities, Springer-Verlag, New York, 1970. MR 43:448

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 33B20, 26D07, 26D15

Retrieve articles in all journals with MSC (1991): 33B20, 26D07, 26D15

Additional Information

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

Keywords: Incomplete gamma function, exponential integral, inequalities
Received by editor(s): May 10, 1995
Received by editor(s) in revised form: April 5, 1996
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society