Converse theorem for the Minkowski inequality

Let (Ω,Σ,μ) a measure space such that 0 < μ(A) < 1 < μ(B) < ∞ for some A, B ∈ Σ. Under some natural conditions on the bijective functions ϕ,ϕ1,ϕ2,ψ,ψ1,ψ2 : (0,∞)→ (0,∞) we prove that if ψ Ω(x+y) ϕ ◦ (x+y)dμ ψ1 Ω(x) ϕ1 ◦ xdμ +ψ2 Ω(y) ϕ2 ◦ ydμ for all nonnegative μ-integrable simple functions x, y : Ω →R (where Ω(x) stands for the support of x, then there exists a real p 1 such that ϕ(t) ϕ(1) = ϕi (t) ϕi (1) =t p, ψ(t) ψ(1) = ψi (t) ψi (1) =t1/p, i = 1, 2. Some generalizations and relevant results for the reversed inequality are also presented.