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Mathematics of Computation

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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
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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?)

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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