Parabolic partial differential equation: Difference between revisions

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In [[mathematics]], '''Levinson's inequality ''' is the following inequality, due to [[Norman Levinson]], involving positive numbers. Let <math>a>0</math> and let <math>f</math> be a given function having a third derivative on the range <math>(0,2a)</math>, and such that
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:<math>f'''(x)\geq 0</math>
 
for all <math>x\in (0,2a)</math>. Suppose <math>0<x_i\leq a</math> for <math> i = 1, \ldots, n</math> and <math>0<p</math>.  Then
 
: <math>\frac{\sum_{i=1}^np_i f(x_i)}{\sum_{i=1}^np_i}-f\left(\frac{\sum_{i=1}^np_ix_i}{\sum_{i=1}^np_i}\right)\le\frac{\sum_{i=1}^np_if(2a-x_i)}{\sum_{i=1}^np_i}-f\left(\frac{\sum_{i=1}^np_i(2a-x_i)}{\sum_{i=1}^np_i}\right).</math>
 
The [[Ky Fan inequality]] is the special case of Levinson's inequality where
 
:<math>p_i=1,\  a=\frac{1}{2},</math>
 
and
 
:<math>f(x)=\log x. \, </math>
 
==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.
 
[[Category:Inequalities]]

Latest revision as of 15:23, 25 September 2014

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