  Let $A$ be a set in an abelian group $G$. For integers $h,r \geq 1$ the
generalized $h$-fold sumset, denoted by $h^{(r)}A$, is the set of sums of $h$
elements of $A$, where each element appears in the sum at most $r$ times. If
$G=\mathbb{Z}$ lower bounds for $|h^{(r)}A|$ are known, as well as the
structure of the sets of integers for which $|h^{(r)}A|$ is minimal. In this
paper we generalize this result by giving a lower bound for $|h^{(r)}A|$ when
$G=\mathbb{Z}/p\mathbb{Z}$ for a prime $p$, and show new proofs for the direct
and inverse problems in $\mathbb{Z}$.
