For a positive integer $k$, let $f(k)$ denote the largest integer $t$ such that for every finite abelian group $G$ and every zero-sum free subset $S$ of $G$, if $|S|=k$ then $|\Sigma(S)|\ge t$. We prove that $f(k) \ge \frac{1}{6}k^2$, which significantly improves a well known result of J.E.~Olson. We also supply some other interesting results on $f(k)$.