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 , 2 a) , and such that f'''(x)\geq 0

for all

x\in ( 0 , 2 a) . Suppose 0 <x _{i\leq} a and 0 <p _{i} for i = 1 , \ldots, n . Then (\sum ^{np} _{i} f(x _{i)})/(- f\left( \sum ^{np} _{i}) (\sum ^{np} _{ix} _{i})/(\right)\le \sum ^{np} _{i}) (\sum ^{np} _{if( 2 a - x} _{i)})/(- f\left( \sum ^{np} _{i}) (\sum ^{np} _{i( 2 a - x} _{i)})/(\sum ^{np} _{i}) \right).

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

p _{i=1,} a = (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.
• Norman Levinson

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