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

:$f\text{'}\text{'}\text{'}(x)\backslash geq\; 0$

for all $x\backslash in\; (0,2a)$. Suppose

: $\backslash frac\{\backslash sum\_\{i=1\}^np\_i\; f(x\_i)\}\{\backslash sum\_\{i=1\}^np\_i\}-f\backslash left(\backslash frac\{\backslash sum\_\{i=1\}^np\_ix\_i\}\{\backslash sum\_\{i=1\}^np\_i\}\backslash right)\backslash le\backslash frac\{\backslash sum\_\{i=1\}^np\_if(2a-x\_i)\}\{\backslash sum\_\{i=1\}^np\_i\}-f\backslash left(\backslash frac\{\backslash sum\_\{i=1\}^np\_i(2a-x\_i)\}\{\backslash sum\_\{i=1\}^np\_i\}\backslash right).$

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

:$p\_i=1,\backslash \; a=\backslash frac\{1\}\{2\},\; \backslash text\{\; and\; \}\; f(x)\; =\; \backslash log\; x.$

==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.

Category:Inequalities (mathematics)