Converse theorem for the Minkowski inequality

Let (Ω,Σ,μ)(Ω,Σ,μ) a measure space such that 0<μ(A)<1<μ(B)<∞0<μ(A)<1<μ(B)<∞ for some A,B∈ΣA,B∈Σ. Under some natural conditions on the bijective functions φ,φ1,φ2,ψ,ψ1,ψ2:(0,∞)→(0,∞)φ,φ1,φ2,ψ,ψ1,ψ2:(0,∞)→(0,∞) we prove that ifψ(∫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→Rx,y:Ω→R (where Ω(x)Ω(x) stands for the support of x, then there exists a real p⩾1p⩾1 such thatφ(t)φ(1)=φi(t)φi(1)=tp,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.