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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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[1] J. T. Chu, On bounds for the normal integral, Biometrika 42 (1955), 263-265. · Zbl 0065.11102
[2] G. M. Fichtenholz, Differential- und Integralrechnung. II, 7th ed., Hochschulbücher für Mathematik [University Books for Mathematics], 62, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978 (German). Translated from the Russian by Brigitte Mai and Walter Mai. · Zbl 0143.27002
[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
[4] D. S. Mitrinović, Analytic inequalities, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165. · Zbl 0199.38101
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