Rejection sampling with inverse-gamma-like density I would like use rejection sampling to sample from a density, $f_y$ on $(0, \infty)$ satisfying
$$f_y(y) \propto \frac{y^{-1}}{1 + y^{-1}}e^{-by^{-1}} $$
I made a first observation that
\begin{align*}
f_y(y) 
& \propto \frac{y^{-1}}{1 + y^{-1}}e^{-by^{-1}} \\
& \leq y^{-1}e^{-by^{-1}} 
\end{align*}
and a second observation that
\begin{align*}
f_y(y) 
& \propto \frac{y^{-1}}{1 + y^{-1}}e^{-by^{-1}} \\
& \leq e^{-by^{-1}} 
\end{align*}
This looks almost like an inverse-gamma$(\alpha, \beta)$ density, except the first parameter is $\alpha = 0$ which is not allowed. The second looks like an exponential distribution, but with a $y^{-1}$ in the exponent instead of just $y$. Could I get a hint on how to proceed?

Both dominating functions are not proportional to densities since they cannot be normalised (they integrate to $+\infty$). The reason why the integral diverges is due in both cases to the upper bound, i.e., when $y$ goes to $+\infty$. However the same applies to $f_y$, which cannot be a density.
