INEQUALITIES OF MINKOWSKI’S TYPE

Abstract

Let $f:[a,b]\to ℝ$ be a nonnegative and nondecreasing function and let $x_i:[a,b]\to ℝ(i=1,\dots ,n)$ be nonnegative and nondecreasing functions with continuous first derivative. If , then

$${\left({\displaystyle {\int }_b^a{\left({\left({\displaystyle \sum _{i=1}^n{x}_i\left(t\right)}\right)}^p\right)}^{\prime }f\left(t\right)dt}\right)}^{1/p}\ge {\displaystyle \sum _{i=1}^n{\left({\displaystyle {\int }_a^b{\left(x_i^p\left(t\right)\right)}^{\prime }f\left(t\right)dt}\right)}^{1/p}}.$$

If $f$ is a nonincreasing function and $x_i(a)=0$ for all $i=1,\dots ,n$, then the reverse inequality is valid.