Let R be a prime ring with characteristic not 2 and σ, τ, α, β, λ, µ, γ automorphisms of R. Let h : R −→ R be a nonzero left(resp.right)-generalized (α, β)−derivation, b ∈ R and V 0 a left (σ, τ)−Lie ideal of R. The main object in this article is to study the situations. (1) h(I) ⊂ C λ,µ (V), (2) bh(I) ⊂ C λ,µ (V) or h(I)b ⊂ C λ,µ (V), (3) hλ(V) = 0, (4) hλ(V)b = 0 or bhλ(V) = 0.