Derivative of renewal density with infinite moment with $\alpha\in(0,1/2]$

Increments of the renewal function related to the distributions with infinite means and regularly varying tails of orders $\alpha\in(0,1]$ were described by Erickson [4,6]. However, explicit asymptotics for the increments are known for $\alpha\in(1/2,1]$ only. For smaller $\alpha$ one can get, generally speaking, only the lower limit of the increments. There are many examples showing that this statement cannot be improved in general. Topchii [1] refine Erikson's results by describing sufficient conditions for regularity of the renewal measure density of the distributions with regularly varying tails with $\alpha\in(0,1/2]$. Here we propose the conditions for regularity of the renewal measure density derivative.