From a result by the present authors and R. R. Janić [Publ. Elektroteh. Fak., Univ. Beogr., Ber. Mat. Fiz. 735-762, 15-18 (1982; Zbl 0547.26012)] several inequalities are derived. For instance: Suppose \(f_ 1,...,f_ m,g_ 1,...,g_{m-1}: X\to {\mathbb{R}}\) are continuous \((X=\times^{r}_{k=1}[a_ k,b_ k]),\quad u_ k:[a_ k,b_ k]\to {\mathbb{R}}\) nondecreasing \((k=1,...,r),\quad f_ 1,...,f_ m\quad are\quad nonnegative,\quad g_ 1,...,g_{m-1}\) positive and \(f_ 1,f_ 2/g_ 1,-g_ 1,...,f_{m-1},f_ m/g_{m-1},-g_{m-1}\) are monotonic in the same sense (each increasing or each decreasing in all variables). Then \[ \int_{X}f_ 1\prod^{m}_{j=1}(f_ j/g_{j-1})du(x_ 1)...du(x_ r)\geq \]
\[ \prod^{m}_{j=1}\int_{X}f_ j du(x_ 1)...du(x_ r)/\prod^{m-1}_{j=1}\int_{X}g_ j \quad du(x_ 1)...du(x_ r). \]