Extended inverse theorems for $h$-fold sumsets in integers

Let $h\ge 2$, $k\ge 5$ be integers and $A$ be a nonempty finite set of $k$ integers. Very recently, Tang and Xing studied extended inverse theorems for $hk-h+1<\mid hA\mid \le hk+2h-3$. In this paper, we extend the work of Tang and Xing and study all possible inverse theorems for $hk-h+1<\mid hA\mid \le hk+3h-4$. Furthermore, we give a range of $\mathrm{\mid }hA\mathrm{\mid }$ for which inverse problems are not possible.