A remark over the converse of Hölder inequality

Abstract

Let be a measure space and be the set of measurable nonnegative real functions defined on . Let be a positive homogenous functional. Suppose that there are two sets , such that and let and be continuous bijective functions of onto . We prove that if there is no positive real number such that and for all , then and must be conjugate power functions. In addition, we prove that if there exists a real number such that then there are nonpower continuous bijective functions and which the above inequality. Also we give an example which shows that the condition that and are continuous functions is essential.

Keywords

• Measure space
• Positive homogenous functional
• Hölder inequality
• Conjugate functions