Levinson's inequality

In mathematics the Levinson's inequality is the following inequality involving positive numbers: Let $a>0$ and $f$ be a given function having a third derivative on the range $]\; 0,2a\; [$, and such that

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

for all $xin\; ]\; 0,2a\; [$. If

:$i=1ldots\; n$ and

we have

$sum\_\{i=1\}^np\_if(x\_i)/sum\_\{i=1\}^np\_i-f\; left\; (sum\_\{i=1\}^np\_ix\_i/sum\_\{i=1\}^np\_i\; ight\; )leq\; sum\_\{i=1\}^np\_if(2a-x\_i)/sum\_\{i=1\}^np\_i-f\; left\; (sum\_\{i=1\}^np\_i(2a-x\_i)/sum\_\{i=1\}^np\_i\; ight\; ).$

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