MathDB

Integral inequality

Source: ISI Entrance 2014, P7

May 11, 2014

calculusintegrationinequalitiesrearrangement inequalityreal analysisIntegral inequality

Problem Statement

Let

f: [0,\infty)\to \mathbb{R}

a non-decreasing function. Then show this inequality holds for all

x,y,z

such that

0\le x<y<z

. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge (z-y)\int_{x}^{z}f(u)\,\mathrm{du} \end{align*}