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On some inequalities for the incomplete gamma function

Author(s): Horst Alzer.
Journal: Math. Comp. 66 (1997), 771-778.
MSC (1991): Primary 33B20; Secondary 26D07, 26D15
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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

$\begin{displaymath}[1-e^{-\beta x^p}]^{1/p}< \frac 1{\Gamma (1+1/p)} \int ^x_0 e^{-t^p} \,dt <[1-e^{-\alpha x^p}]^{1/p}\end{displaymath}$

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

$\begin{displaymath}-\log (1-e^{-ax})\le \int ^\infty _x \frac {e^{-t}}t\,dt\le -\log (1-e^{-bx})\end{displaymath}$

hold for all $x>0$ .

References:

1.: J. T. Chu, On bounds for the normal integral, Biometrika 42 (1955), 263-265. MR 16:838f
2.: G. M. Fichtenholz, Differential- und Integralrechnung, II, Dt. Verlag Wissensch., Berlin, 1979. MR 80f:26001
3.: W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys. 38 (1959), 77-81. MR 21:2067
4.: D. S. Mitrinovi\'{c}, Analytic inequalities, Springer-Verlag, New York, 1970. MR 43:448


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