Subgraphs with a positive minimum semidegree in digraphs with large outdegree

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