Some properties on the Schur multiplier of a pair of groups

J.L. Loday (1978) introduced the concept of relative central extension and G. Ellis (1998) gave the notion of covering pair of a pair of groups (G,N). In this paper under some condition we show the existence of covering pair for the pair of groups (G,N) and also show that every relative central extension is a homomorphic image of a covering pair of (G,N). Finally, we present some inequalities for the Schur multiplier of a pair of finite groups (G,N).