Directed graphs without rainbow stars

In a rainbow version of the classical Turán problem one considers multiple graphs on a common vertex set, thinking of each graph as edges in a distinct color, and wants to determine the minimum number of edges in each color, or their sum, which guarantees existence of a rainbow copy (having each edge from a different graph) of a given graph. Here, we find an optimal solution for this problem, both for the minimum and the sum, for any directed star and any number of colors.