Subgraphs with a positive minimum semidegree in digraphs with large outdegree

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