  Let $D=(V,A)$ be a digraph of order $n$, $S$ a subset of $V$ of size $k$ and
$2\le k\leq n$. Strong subgraphs $D_1, \dots , D_p$ containing $S$ are said to
be internally disjoint if $V(D_i)\cap V(D_j)=S$ and $A(D_i)\cap
A(D_j)=\emptyset$ for all $1\le i<j\le p$. Let $\kappa_S(D)$ be the maximum
number of internally disjoint strong digraphs containing $S$ in $D$. The strong
subgraph $k$-connectivity is defined as $$\kappa_k(D)=\min\{\kappa_S(D)\mid
S\subseteq V, |S|=k\}.$$ A digraph $D=(V, A)$ is called minimally strong
subgraph $(k,\ell)$-connected if $\kappa_k(D)\geq \ell$ but for any arc $e\in
A$, $\kappa_k(D-e)\leq \ell-1$. In this paper, we first give a sharp upper
bound for the parameter $\kappa_k(D)$ and then study the minimally strong
subgraph $(k,\ell)$-connected digraphs.
