Optimally large cuts in both directions

We prove that for every integer $d\ge2$ and all sufficiently large $m$, every digraph with $m$ edges and minimum outdegree at least $d$ has a vertex bipartition for which each directed cut is of size at least \[\left\lceil\frac{d-1}{2(2d-1)}m+\frac{d^2}{2(2d-1)}\right\rceil.\] The bound is sharp for