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 $$h_{f}(g)=sum_{uin v(g)}f(d_{u}).$$ 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 $left[ 0,+infty right)$.