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 \(x> 0\).


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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