We prove several numerical radius inequalities for products of two Hilbert space operators. Some of our inequalities improve well-known ones. More precisely, we prove that, if A, B ∈ B(H) such that A is self-adjoint with λ 1 = min λ i ∈ σ (A) (the spectrum of A) and λ 2 = max λ i ∈ σ (A). Then ω(AB) ≤ Aω(B) + A − |λ 1 + λ 2 | 2 D B where D B = inf λ∈C B − λI. In particular, if A > 0 and σ(A) ⊆ [kA, A], then ω(AB) ≤ (2 − k)A ω(B).