Vertex-degree function index on tournaments

Let G be a simple graph with vertex set V = V (G) and edge set E = E(G). For a real function f defined on nonnegative real numbers, the vertex-degree function index H-f(G) is defined as H-f(G) = Sigma(u is an element of V(G)) f (du). In this paper we introduce the vertex-degree function index H-f (D) of a digraph D. After giving some examples and basic properties of H-f(D), we find the extremal values of H-f among all tournaments with a fixed number of vertices, when f is a continuous and convex (or concave) real function on [0, +infinity).