Abstract
Let (Ω,𝒜,μ) be a measure space and ℒ be the set of measurable nonnegative real functions defined on Ω. Let F:ℒ→[0,∞] be a positive homogenous functional. Suppose that there are two sets A, B∈𝒜 such that 0<F(χA)<1<F(χB)<∞ and let ϕ and ψ be continuous bijective functions of [0,∞) onto [0,∞). We prove that if there is no positive real number d such that {F(χC):C∈𝒜,F(χC)>0}⊂{dk:k∈Z} and
F(xy)≤ϕ−1(F(ϕ∘x))ψ−1(F(ψ∘y))
for all x,y∈{αχC∈ℒ:F(χC)<∞,α∈R}, then ϕ and ψ must be conjugate power functions.
In addition, we prove that if there exists a real number d>0 such that {F(χC):C∈𝒜,F(χC)>0}⊂{dk:k∈Z} then there are nonpower continuous bijective functions ϕ and ψ which the above inequality. Also we give an example which shows that the condition that ϕ and ψ are continuous functions is essential.