Exact lower and upper bounds on the incomplete gamma function

Abstract

Lower and upper bounds $B_a(x)$ on the incomplete gamma function $\Gamma (a,x)$ are given for all real $a$ and all real $x>0$. These bounds $B_a(x)$ are exact in the sense that $B_a(x)\underset{x\downarrow 0}{\sim }\Gamma (a,x)$ and $B_a(x)\underset{x\to \infty }{\sim }\Gamma (a,x)$. Moreover, the relative errors of these bounds are rather small for other values of $x$, away from $0$ and $\infty$.

arXiv commentTo appear in Mathematical Inequalities & Applications