Subhomogeneity and subadditivity of the LP-norm like functionals

For a measure space (Omega, Sigma, mu) denote by S = S(Omega, Sigma, mu) the set of all mu-integrable simple functions x : Omega -> R. For a bijection phi : (0, infinity) -> (0, infinity) we consider the functional P phi : S -> [0, infinity), P phi(x) := phi(-1) (integral(Omega(x)) phi o broken vertical bar x broken vertical bar d mu) where Omega(x) is the support of x is an element of S. One of the results says that if the measure mu, has values in (0, 1) and in (1, infinity), the function phi is monotonic and P-phi satisfies the inequality P phi(tx) <= tP(phi)(x), t > 1, x is an element of S, then yo is a power function. Some characterizations of the functions phi in two remaining cases when either mu(Sigma)boolean AND(1, infinity) = theta or mu (0, 1) = theta are given. The subadditivity of P-phi, i.e. a generalization of the Minkowski inequality, is also considered. (C) 2013 Elsevier Inc. All rights reserved.