Concentration probabilities for distributions with $\exp(-c t^\alpha)$ tails

Question

I am trying to generalize the concentration inequalities for the sub-exponential and sub-gaussian distributions to a larger family with tail $\exp(-c t^\alpha)$ for arbitrary $\alpha > 0$. In particular, we have the Hoeffding inequality for sub-gaussian distributions (corresponding to $\alpha = 2$) and the Bernstein inequality for the sub-exponential distributions (corresponding to $\alpha = 1$). However, both the proofs I have read heavily rely on the specific $\alpha$ values and are hard to generalize. Is there any way to generalize these inequalities to arbitrary $\alpha > 0$?