Subhomogeneity and subadditivity of the LpLp-norm like functionals

For a measure space (Ω,Σ,μ)(Ω,Σ,μ) denote by S=S(Ω,Σ,μ)S=S(Ω,Σ,μ) the set of all μμ-integrable simple functions x:Ω→Rx:Ω→R. For a bijection φ:(0,∞)→(0,∞)φ:(0,∞)→(0,∞) we consider the functional Pφ:S→[0,∞), Pφ(x):=φ−1(∫Ω(x)φ∘|x|dμ), where Ω(x)Ω(x) is the support of x∈Sx∈S. One of the results says that if the measure μμ has values in (0,1)(0,1) and in (1,∞)(1,∞), the function φφ is monotonic and Pφ satisfies the inequality Pφ(tx)≤tPφ(x),t>1,x∈S, then φφ is a power function. Some characterizations of the functions φφ in two remaining cases when either μ(Σ)∩(1,∞)=0̸μ(Σ)∩(1,∞)=0̸ or μ(Σ)∩(0,1)=0̸μ(Σ)∩(0,1)=0̸ are given. The subadditivity of Pφ, i.e. a generalization of the Minkowski inequality, is also considered.