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

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

${\left[1-{e}^{-\beta {x}^{p}}\right]}^{1/p}<\frac{1}{{\Gamma }\left(1+1/p\right)}{\int }_{0}^{x}{e}^{-{t}^{p}}dt<{\left[1-{e}^{-\alpha {x}^{p}}\right]}^{1/p}$

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

$-log\left(1-{e}^{-ax}\right)\le {\int }_{x}^{\infty }\frac{{e}^{-t}}{t}dt\le log\left(1-{e}^{-bx}\right)$

hold for all $x>0$.

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