signed total roman k-domination in directed graphs

let $d$ be a finite and simple digraph with vertex set $v(d)$‎.‎a signed total roman $k$-dominating function (str$k$df) on‎‎$d$ is a function $f:v(d)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin n^{-}(v)}f(x)ge k$ for each‎‎$vin v(d)$‎, ‎where $n^{-}(v)$ consists of all vertices of $d$ from‎‎which arcs go into $v$‎, ‎and (ii) every vertex $u$ for which‎‎$f(u)=-1$ has an inner neighbor $v$ for which $f(v)=2$‎.‎the weight of an str$k$df $f$ is $omega(f)=sum_{vin v (d)}f(v)$‎.‎the signed total roman $k$-domination number $gamma^{k}_{str}(d)$‎‎of $d$ is the minimum weight of an str$k$df on $d$‎. ‎in this paper we‎‎initiate the study of the signed total roman $k$-domination number‎‎of digraphs‎, ‎and we present different bounds on $gamma^{k}_{str}(d)$‎.‎in addition‎, ‎we determine the signed total roman $k$-domination‎‎number of some classes of digraphs‎. ‎some of our results are extensions‎‎of known properties of the signed total roman $k$-domination‎‎number $gamma^{k}_{str}(g)$ of graphs $g$‎.