Minimum feedback arc set of m-free digraphs

For a simple digraph G, let $\beta (G)$ be the size of the smallest subset $X\subseteq E(G)$ such that $G-X$ has no directed cycles, and let $\gamma (G)$ be the number of unordered pairs of nonadjacent vertices in G. A digraph G is called m‐free if G has no directed cycles of length at most m. This paper proves that $\beta (G)⩽\frac{1}{m-2}\gamma (G)$ for any m‐free digraph G, which generalizes some known results.