The Stanley regularity of complete intersections and ideals of mixed products

Let S be a polynomial ring over a field k and L ⊆ S a monomial ideal. We give some inequalities on Stanley regularity of monomial ideals. As consequences, we prove that sregS(S/L) ≤regS(S/L) and sregS(L) ≤regS(L) hold in the following cases: (1) L is a complete intersection; (2) L is an ideal of mixed products.