On some inequalities for the incomplete gamma function
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- Math. Comp. 66 (1997), 771-778 Request permission
Abstract:
Let $p\ne 1$ 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 \[ -\log (1-e^{-ax})\le \int ^\infty _x \frac {e^{-t}}t dt\le -\log (1-e^{-bx})\] hold for all $x>0$.References
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- G. M. Fichtenholz, Differential- und Integralrechnung. II, 7th ed., Hochschulbücher für Mathematik [University Books for Mathematics], vol. 62, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978 (German). Translated from the Russian by Brigitte Mai and Walter Mai. MR 524565
- Walter Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. and Phys. 38 (1959/60), 77–81. MR 103289, DOI 10.1002/sapm195938177
- D. S. Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. MR 0274686, DOI 10.1007/978-3-642-99970-3
Additional Information
- Horst Alzer
- Affiliation: Morsbacher Str. 10, 51545 Waldbröl, Germany
- MR Author ID: 238846
- Received by editor(s): May 10, 1995
- Received by editor(s) in revised form: April 5, 1996
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp. 66 (1997), 771-778
- MSC (1991): Primary 33B20; Secondary 26D07, 26D15
- DOI: https://doi.org/10.1090/S0025-5718-97-00814-4
- MathSciNet review: 1397438