Subgraphs with a positive minimum semidegree in digraphs with large outdegree

We prove that every $n$-vertex directed graph $G$ with the minimum outdegree ${\delta }^{+}(G)=d$ contains a subgraph $H$ satisfying $$\min \{{\delta }^{+}(H),{\delta }^{-}(H)\}\ge \frac{d(d+1)}{2n}\mathrm{.}$$ We also show that if $d=o(n)$ then this bound is asymptotically best possible.