On cross-intersecting families of sets

A family ℓ-element subsets and a family κ-element subsets of an n-element set are cross-intersecting if every set from A has a nonempty intersection with every set from B. We compare two previously established inequalities each related to the maximization of the product |A||ℬ|, and give a new and short proof for one of them. We also determine the maximum of |A|ω_l + |ℬ|ω_k for arbitrary positive weights ω_ℓ,ω _κ .