The Pexider type generalization of the Minkowski inequality

Let ( 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 ) d μ ) ≤ ψ − 1 ( ∫ Ω ( x ) ψ ∘ x d μ ) + γ − 1 ( ∫ Ω ( y ) γ ∘ y d μ ) 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 ) = t p . 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 ) < 1 , then there are some broad classes of non-power functions which satisfy the above Minkowski type inequality. Moreover we prove that, in the converse of the Minkowski inequality theorem, the assumption of the increasing monotonicity of φ is essential.