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

${\displaystyle f'''(x)\geq 0}$

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

${\displaystyle {\frac {\sum _{i=1}^{n}p_{i}f(x_{i})}{\sum _{i=1}^{n}p_{i}}}-f\left({\frac {\sum _{i=1}^{n}p_{i}x_{i}}{\sum _{i=1}^{n}p_{i}}}\right)\leq {\frac {\sum _{i=1}^{n}p_{i}f(2a-x_{i})}{\sum _{i=1}^{n}p_{i}}}-f\left({\frac {\sum _{i=1}^{n}p_{i}(2a-x_{i})}{\sum _{i=1}^{n}p_{i}}}\right).}$

The Ky Fan inequality is the special case of Levinson's inequality, where

${\displaystyle p_{i}=1,\ a={\frac {1}{2}},{\text{ and }}f(x)=\log x.}$