  Given a positive integer $m\ge 3$, let $ch(m)$ be the smallest positive
constant with the following property:
  \emph{ Every simple directed graph on $n\ge 3$ vertices all whose outdegrees
are at least $ch(m)\cdot n$ contains a directed cycle of length at most $m$.}
  Caccetta and H\"{a}ggkvist conjectured that $ch(m)=1/m$, which if true, would
be the best possible. In this paper, we prove the following result:
  \emph{ For every integer $m\ge 3$, let $\alpha(m)$ be the unique real root in
$(0,1)$ of the equation} \begin{equation*} (1-x)^{m-2}=\frac{3x}{2-x}.
\end{equation*} Then $ch(m)\le \alpha(m)$.
  This generalizes results of Shen who proved that $ch(3)\le
3-\sqrt{7}<0.35425$, and Liang and Xu who showed that $ch(4)< 0.28866$ and
$ch(5)<0.24817$.
  We then slightly improve the above inequality by using the minimum feedback
arc set approach initiated by Chudnovsky, Seymour, and Sullivan. This results
in extensions of the findings of Hamburger, Haxell and Kostochka (in the case
$m=3$), and Liang and Xu (in the case $m=4$).
