## On some inequalities for the incomplete gamma function.(English)Zbl 0865.33002

Summary: Let $$p\neq 1$$ 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:

 33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) 26D07 Inequalities involving other types of functions 26D15 Inequalities for sums, series and integrals
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### References:

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