Levinson's inequality

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

$f'''(x)\geq 0$

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

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

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

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

References

• Scott Lawrence and Daniel Segalman: A generalization of two inequalities involving means, Proceedings of the American Mathematical Society. Vol 35 No. 1, September 1972.
• Norman Levinson: Generalization of an inequality of Ky Fan, Journal of Mathematical Analysis and Applications. Vol 8 (1964), 133–134.