On supremum operators

The set of all nonnegative measurable functions and its subset consisting of all nonincreasing functions are denoted to study the supremum operators. The inequality is characterized for the Hardy operator, in which the constant C is minimal among all possible constants. The results shows that for a continuous function, the inequality for certain conditions. For a jointly measurable nonnegative function, the supremum operator is defined. The necessary and sufficient for the inequality with nonnegative weight functions are also defined.