The Pexider type generalization of the Minkowski inequality

AbstractLet (Ω,Σ,μ) be a measure space such that 0<μ(A)<1<μ(B)<∞ for some A,B∈Σ. The following converse Minkowski inequality theorem is proved in Matkowski (2008) [4]. If φ,ψ,γ:(0,∞)→(0,∞) are bijective, φ is increasing, and φ−1(∫Ω(x+y)φ∘(x+y)dμ)≤ψ−1(∫Ω(x)ψ∘xdμ)+γ−1(∫Ω(y)γ∘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)=ψ(t)ψ(1)=γ(t)γ(1)=tp. In the present paper we show that if, in the basic measure space, there is no A∈Σ such that either 1<μ(A)<∞ or 0<μ(A)