A nonlinear bound for the number of subsequence sums

We show that a finite zero-sum-free sequence $\alpha$ over an abelian group has at least $c|\alpha|^{4/3}$ distinct subsequence sums, unless $\alpha$ is "controlled" by a small number of its terms; here $|\alpha|$ denotes the number of terms of $\alpha$, and $c>0$ is an absolute constant.