Subhomogeneity and subadditivity of the -norm like functionals

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