On weighted multidimensional embeddings for monotone functions

We characterize the inequality 7817 (ₑ₍_+ fq u) ^1/q C (ₑ₍_+ fᵖ v) ^1/p, \, \, 0<q, p <, 7817 for monotone functions f 0 and nonnegative weights u and v. The case q < p is new and the case 0<p q < is extended to a modular inequality with N- functions. A remarkable fact concerning the calculation of C is pointed out.