Levinson's inequality explained

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)\geq0

for all

x\in(0,2a)

. Suppose

0<xi\leqa

and

0<pi

for

i=1,\ldots,n

. Then
np
\sumf(xi)
i
-f\left(
np
\sum
i
np
\sumi
ix
\right)\le
np
\sum
i
np
\sumi)
if(2a-x
-f\left(
np
\sum
i
np
\sumi)
i(2a-x
np
\sum
i

\right).

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

pi=1,a=

1
2

,andf(x)=logx.

References

Generalization of an inequality of Ky Fan, Journal of Mathematical Analysis and Applications. Vol 8 (1964), 133–134.