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 $$\min\left\{\delta^+(H), \delta^-(H) \right\} \ge \frac{d(d+1)}{2n} \,.$$ We also show that if $d = o(n)$ then this bound is asymptotically best possible.