Journal of Inequalities and Applications
Volume 3 (1999), Issue 2, Pages 127-135
doi:10.1155/S1025583499000089
A remark over the converse of Hölder inequality
Departamento de Matemática, Fac. de Ciencias Ex. F.Q.y N, Universidad Nacional de Río Cuarto, Río Cuarto 5800, Argentina
Received 28 October 1997; Revised 16 February 1998
Copyright © 1999 Héctor H. Cuenya and Fabián E. Levis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.