Motivated by a problem of Lévy-flights in a potential, I am looking for the asymptotics of a function $f(y)$, which obeys the the integro-differential equation $$ e^{2y} f(y) = C \frac{d}{dy}\int_y^\infty \frac{f(z)}{(z-y)^{\alpha-1}} dy,$$ with a constant $C$, and $\alpha\in(1,2)$ in the limit $y\to \infty$. I know from numerical solutions that it decays faster than exponentially, but I struggled to get an analytical expression.
EDIT: I am assuming $f$ to be positive (it is a probability density in the original problem) and it decays to zero at infinity faster than exponentially.