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

$f^{‴}(x)\ge 0$

for all $x\in (0,2a)$. Suppose $0<x_i\le a$ and $0<p_i$ for $i=1,\dots ,n$. Then

$\frac{\sum _{i=1}^n{p}_{i}f(x_i)}{\sum _{i=1}^n{p}_i}-f\left(\frac{\sum _{i=1}^n{p}_i{x}_i}{\sum _{i=1}^n{p}_i}\right)\le \frac{\sum _{i=1}^n{p}_{i}f(2a-x_i)}{\sum _{i=1}^n{p}_i}-f\left(\frac{\sum _{i=1}^n{p}_i(2a-x_i)}{\sum _{i=1}^n{p}_i}\right).$

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

$$\begin{gathered} p_i=1, \\ a=\frac{1}{2},\text{ and }f(x)=\log x. \end{gathered}$$

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.