  Let $G$ be a directed planar graph on $n$ vertices, with no directed cycle of
length less than $g\ge 4$. We prove that $G$ contains a set $X$ of vertices
such that $G-X$ has no directed cycle, and $|X|\le \tfrac{5n-5}9$ if $g=4$,
$|X|\le \tfrac{2n-5}4$ if $g=5$, and $|X|\le \tfrac{2n-6}{g}$ if $g\ge 6$. This
improves recent results of Golowich and Rolnick.
