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

Alzer, Horst

Math. Comput. 66, No. 218, 771-778 (1997).

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

Keywords:

incomplete gamma function; exponential integral; inequalities

References:

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[3]	Walter Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. and Phys. 38 (1959/60), 77 – 81. · Zbl 0094.04104
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