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 $0<x_i\leq a$ for $i = 1, \ldots, n$ and $0<p$. Then

$\frac{\sum_{i=1}^np_i f(x_i)}{\sum_{i=1}^np_i}-f\left(\frac{\sum_{i=1}^np_ix_i}{\sum_{i=1}^np_i}\right)\le\frac{\sum_{i=1}^np_if(2a-x_i)}{\sum_{i=1}^np_i}-f\left(\frac{\sum_{i=1}^np_i(2a-x_i)}{\sum_{i=1}^np_i}\right).$

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

$p_i=1,\ a=\frac{1}{2},$

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.