Summary: Weight characterizations of weighted modular inequalities for operators on the cone of monotone functions are given in terms of composition operators on arbitrary nonnegative functions with changes in weights. The results extend to modular inequalities, those corresponding to weighted Lebesgue spaces given by E. T. Sawyer
[Stud. Math. 96, No. 2, 145-158 (1990; Zbl 0705.42014
)]. Application to Hardy and fractional integral operators on monotone functions are given.