Levinson's inequality

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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 $]0,2 a [$ , and such that

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

for all $x\in ]0,2a[$ . If $0<x_i\leq a$ for every

$i=1\ldots n$ and

0 < p

we have

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

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

[edit] 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