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On some inequalities for the incomplete gamma function

Authors:	Horst Alzer

Institution:	Morsbacher Str. 10, 51545 Waldbröl, Germany

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 . And, we determine all real numbers and 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 .

Keywords:	Incomplete gamma function exponential integral inequalities

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