Levinson's inequality

In mathematics, Levinson's inequality is the following inequality, due to Norman Levinson, involving positive numbers. Let [math]\displaystyle{ a\gt 0 }[/math] and let [math]\displaystyle{ f }[/math] be a given function having a third derivative on the range [math]\displaystyle{ (0,2a) }[/math], and such that

[math]\displaystyle{ f'''(x)\geq 0 }[/math]

for all [math]\displaystyle{ x\in (0,2a) }[/math]. Suppose [math]\displaystyle{ 0\lt x_i\leq a }[/math] and [math]\displaystyle{ 0\lt p_i }[/math] for [math]\displaystyle{ i = 1, \ldots, n }[/math]. Then

[math]\displaystyle{ \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]\displaystyle{ p_i=1,\ a=\frac{1}{2}, \text{ and } 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.