  An $(r,s)$-directed hypergraph is a directed hypergraph with $r$ vertices in
tail and $s$ vertices in head of each arc. Let $G$ be an $(r,s)$-directed
hypergraph. For any real numbers $p$, $q\geq 1$, we define the $(p,q)$-spectral
radius $\lambda_{p,q}(G)$ as \[ \lambda_{p,q}(G):=\max_{||{\bf x}||_p=||{\bf
y}||_q=1} \sum_{e\in E(G)}\Bigg(\prod_{u\in T(e)}x_u\Bigg)\Bigg(\prod_{v\in
H(e)}y_v\Bigg), \] where ${\bf x}=(x_1, \ldots, x_m)^{{\rm T}}$, ${\bf
y}=(y_1,\ldots, y_n)^{{\rm T}}$ are real vectors; and $T(e)$, $H(e)$ are the
tail and head of arc $e$, respectively. We study some properties about
$\lambda_{p,q}(G)$ including the bounds and the spectral relation between $G$
and its components.
  The $\alpha$-normal labeling method for uniform hypergraphs was introduced by
Lu and Man in 2014. It is an effective method in studying the spectral radii of
uniform hypergraphs. In this paper, we develop the $\alpha$-normal labeling
method for calculating the $(p,q)$-spectral radii of $(r,s)$-directed
hypergraphs. Finally, some applications of $\alpha$-normal labeling method are
given.
