A bound on the genus of a curve with Cartier operator of small rank

Ekedahl showed that the genus of a curve in characteristic $p>0$ with zero Cartier operator is bounded by $p(p-1)\mathrm{/}2$. We show the bound $p+p(p-1)\mathrm{/}2$ in case the rank of the Cartier operator is 1, improving a result of Re.