Levinson's inequality

In mathematics, Levinson's inequality is the following inequality, due to Norman Levinson, involving positive numbers. Let ${\displaystyle a>0}$ and let ${\displaystyle f}$ be a given function having a third derivative on the range ${\displaystyle (0,2a)}$, and such that

for all ${\displaystyle x\in (0,2a)}$. Suppose ${\displaystyle 0<x_{i}\leq a}$ for ${\displaystyle i=1,\ldots ,n}$ and ${\displaystyle 0<p}$. Then

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