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Zbl 0865.33002
Alzer, Horst
On some inequalities for the incomplete gamma function. (English)
[J] Math. Comput. 66, No.218, 771-778 (1997). ISSN 0025-5718; ISSN 1088-6842/e

Summary: 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 $$[1- e^{-\beta x^p}]^{1/p}<\frac{1}{\Gamma(1+ 1/p)}\int^x_0 e^{-t^p}dt<[1-e^{-\alpha x^p}]^{1/p}$$ are valid for all $x>0$. And, we determine all real numbers $a$ and $b$ such that $$-\log(1- e^{-ax})\leq \int^\infty_x \frac{e^{-t}}{t} dt\leq \log(1- e^{-bx})$$ hold for all $x> 0$.

MSC 2000:: *33B20 Incomplete beta and gamma functions
26D07 Inequalities involving other types of real functions
26D15 Inequalities for sums, series and integrals of real functions

Keywords: incomplete gamma function; exponential integral; inequalities

Cited in: Zbl 0916.33001

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