A sharp lower bound for $h$-fold restricted sumsets of integers

Let $A\subseteq [0,L]$ be a set of $k⩾9$ integers such that $0,L\in A$ and $\gcd (A)=1$. Define the $h$-fold restricted sumset $$h^{\wedge }A=\{a_1+\cdots +a_h:a_i\in A,1⩽i⩽h,a_i\mathrm{\ne }{a}_j\mathrm{f}\mathrm{o}\mathrm{r}i\mathrm{\ne }j\}\mathrm{.}$$ In this paper, we prove that, for every $1⩽h⩽k-1$, we have $$\mathrm{\mid }{h}^{\wedge }A\mathrm{\mid }⩾h(k-h)-k+2+\min \{L,h(k-h)-{\mathbf{1}}_h\},$$ where ${\mathbf{1}}_h=1$ for $h\in \{2,k-2\}$ and ${\mathbf{1}}_h=0$ otherwise. The bound is sharp for every $1⩽h⩽k-1$ and the cardinality threshold $k⩾9$ cannot be decreased. We also give a lower bound for the union of two consecutive restricted sumsets in a cyclic group and characterize equality.