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 ]0,2a[, 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) = logx.

[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
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