  Let $D$ be an arc-colored digraph. The arc number $a(D)$ of $D$ is defined as
the number of arcs of $D$. The color number $c(D)$ of $D$ is defined as the
number of colors assigned to the arcs of $D$. A rainbow triangle in $D$ is a
directed triangle in which every pair of arcs have distinct colors. Let $f(D)$
be the smallest integer such that if $c(D)\geq f(D)$, then $D$ contains a
rainbow triangle. In this paper we obtain $f(\overleftrightarrow{K}_{n})$ and
$f(T_n)$, where $\overleftrightarrow{K}_{n}$ is a complete digraph of order $n$
and $T_n$ is a strongly connected tournament of order $n$. Moreover we
characterize the arc-colored complete digraph $\overleftrightarrow{K}_{n}$ with
$c(\overleftrightarrow{K}_{n})=f(\overleftrightarrow{K}_{n})-1$ and containing
no rainbow triangles. We also prove that an arc-colored digraph $D$ on $n$
vertices contains a rainbow triangle when $a(D)+c(D)\geq
a(\overleftrightarrow{K}_{n})+f(\overleftrightarrow{K}_{n})$, which is a
directed extension of the undirected case.
