A vertex subset \(S\) of a digraph \(D = (V, A)\) is called an out-dominating (resp.,in-dominating) set of \(D\) if every vertex in \(V – S\) is adjacent from (resp., to) some vertex in \(S\). The out-domination (resp., in-domination) number of \(D\), denoted by \(\gamma^+(D)\) (resp.,\(\gamma^-(D)\)), is the minimum cardinality of an out-dominating (resp., in-dominating) set of \(D\). In 1999, Chartrand et al. proved that \(\gamma^+(D) + \gamma^-(D) \leq \frac{4n}{3}\) for every digraph \(D\) of order \(n\) with no isolated vertices. In this paper, we determine the values of \(\gamma^+(D) + \gamma^-(D)\) for rooted trees and connected contrafunctional digraphs \(D\), based on which we show that \(\gamma^+(D) + \gamma^-(D) \leq \frac{(2k+2)n}{2k+1}\) for every digraph \(D\) of order \(n\) with minimum out-degree or in-degree no less than \(1\), where \(2k + 1\) is the length of a shortest odd directed cycle in \(D\). Our result partially improves the result of Chartrand et al. In particular, if \(D\) contains no odd directed cycles, then \(\gamma^+(D) + \gamma^-(D) \leq n\).