## Abstract

Let *T* (*X* ∪ *Y, A*) be a bipartite tournament with partite sets *X, Y* and arc set *A*. For any vertex *x* ∈ *X* ∪*Y*, the second out-neighbourhood *N*
^{++}(*x*) of *x* is the set of all vertices with distance 2 from *x*. In this paper, we prove that *T* contains at least two vertices *x* such that |*N*
^{++}(*x*)| ≥ |*N*
^{+}(*x*)| unless *T* is in a special class ℬ_{1} of bipartite tournaments; show that *T* contains at least a vertex *x* such that |*N*
^{++}(*x*)| ≥ |*N*
^{−}(*x*)| and characterize the class ℬ_{2} of bipartite tournaments in which there exists exactly one vertex *x* with this property; and prove that if |*X*| = |*Y* | or |*X*| ≥ 4|*Y* |, then the bipartite tournament *T* contains a vertex *x* such that |*N*
^{++}(*x*)|+|*N*
^{+}(*x*)| ≥ 2|*N*
^{−}(*x*)|.