On a separation for the Cauchy equation on spheres

We give sufficient conditions for separating functions p and q, satisfying for a given function γ the conditional inequalities γ(x)=γ(y)⟹p(x+y)≤p(x)+p(y),γ(x)=γ(y)⟹q(x+y)≥q(x)+q(y), by an additive mapping.