Truncated gamma rejection sampling with a Pareto envelope ($x_{\ast }=(\alpha +b)/\beta$)

For target density proportional to $x^{\alpha -1}{e}^{-\beta x}$ on $x>a>0$, use the Pareto distribution proposal $g(x)=b{a}^b{x}^{-b-1}$ with $b>0$. Its density ratio is proportional to $x^{\alpha +b}{e}^{-\beta x}$, maximized at $(\alpha +b)/\beta$ if that point exceeds $a$. The tight envelope yields the highest acceptance probability for this fixed proposal. An inverse draw is $a{U}^{-1/b}$ with $U$ uniformly distributed on $(0,1)$.