Levinson's inequality - Alchetron, The Free Social Encyclopedia

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 ) ≥ 0

for all x ∈ ( 0 , 2 a ) . Suppose 0 < x i ≤ a for i = 1 , … , n and 0 < p . Then

∑ i = 1 n p i f ( x i ) ∑ i = 1 n p i − f ( ∑ i = 1 n p i x i ∑ i = 1 n p i ) ≤ ∑ i = 1 n p i f ( 2 a − x i ) ∑ i = 1 n p i − f ( ∑ i = 1 n p i ( 2 a − x i ) ∑ i = 1 n p i ) .

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

Levinson's inequality Wikipedia

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