The numerical range of matrix products

We discuss what can be said about the numerical range of the matrix product A₁A₂ when the numerical ranges of A₁ and A₂ are known. If two compact convex subsets K₁, K₂ of the complex plane are given, we discuss the issue of finding a compact convex subset K such that whenever Aⱼ ($j=1,2$) are either unrestricted matrices or normal matrices of the same shape with W(Aⱼ) ⊆ Kⱼ, it follows that W(A₁A₂) ⊆ K. We do this by defining specific deviation quantities for both the unrestricted case and the normal case.