In mathematics, Levinson's inequality is the following inequality, due to Norman Levinson, involving positive numbers. Let and let be a given function having a third derivative on the range , and such that
for all . Suppose and for . Then
The Ky Fan inequality is the special case of Levinson's inequality, where
Referencesedit
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.
January 01, 1970
levinson, inequality, mathematics, following, inequality, norman, levinson, involving, positive, numbers, displaystyle, displaystyle, given, function, having, third, derivative, range, displaystyle, such, that, displaystyle, displaystyle, suppose, displaystyle. In mathematics Levinson s inequality is the following inequality due to Norman Levinson involving positive numbers Let a gt 0 displaystyle a gt 0 and let f displaystyle f be a given function having a third derivative on the range 0 2 a displaystyle 0 2a and such that f x 0 displaystyle f x geq 0 for all x 0 2 a displaystyle x in 0 2a Suppose 0 lt x i a displaystyle 0 lt x i leq a and 0 lt p i displaystyle 0 lt p i for i 1 n displaystyle i 1 ldots n 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 displaystyle frac sum i 1 n p i f x i sum i 1 n p i f left frac sum i 1 n p i x i sum i 1 n p i right leq frac sum i 1 n p i f 2a x i sum i 1 n p i f left frac sum i 1 n p i 2a x i sum i 1 n p 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 displaystyle p i 1 a frac 1 2 text and f x log x References editScott 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 Retrieved from https en wikipedia org w index php title Levinson 27s inequality amp oldid 1091334237, wikipedia, wiki, book, books, library,