On some inequalities for the incomplete gamma function

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 $x>0$.