On some inequalities for the incomplete gamma function

Source abstract

Let p ≠ 1 p\ne 1 be a positive real number. We determine all real numbers α = α ( p ) \alpha = \alpha (p) and β = β ( p ) \beta =\beta (p) such that the inequalities $$[ 1 − e − β x p ] 1 / p > 1 Γ ( 1 + 1 / p ) ∫ 0 x e − t p d t > [ 1 − e − α x p ] 1 / p [1-e^{-\beta x^p}]^{1/p}> \frac 1{\Gamma (1+1/p)} \int ^x_0 e^{-t^p} \,dt >[1-e^{-\alpha x^p}]^{1/p}$$ are valid for all x > 0 x>0 . And, we determine all real numbers a a and b b such that $$− log ⁡ ( 1 − e − a x ) ≤ ∫ x ∞ e − t t d t ≤ − log ⁡ ( 1 − e − b x ) -\log (1-e^{-ax})\le \int ^\infty _x \frac {e^{-t}}t\,dt\le -\log (1-e^{-bx})$$ hold for all x > 0 x>0 .