Second neighborhood via first neigh borhood in digraph

Let D be a simple digraph without loops or digons. For any v∈V(D)
, the first out-neighborhood N +(v) is the set of all vertices with out-distance 1 from v and the second neighborhood N ++(v) of v is the set of all vertices with out-distance 2 from v. We show that every simple digraph without loops or digons contains a vertex v such that |N++(v)|≥γ|N+(v)|, where γ = 0.657298... is the unique real root of the equation 2x 3 + x 2 -1 = 0.