Levinson's inequality

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 $$(0,2a)$$, and such that


 * $$f'''(x)\geq 0$$

for all $$x\in (0,2a)$$. Suppose $$0<x_i\leq a$$ and $$0<p_i$$ for $$ i = 1, \ldots, n$$. Then


 * $$\frac{\sum_{i=1}^np_i f(x_i)}{\sum_{i=1}^np_i}-f\left(\frac{\sum_{i=1}^np_ix_i}{\sum_{i=1}^np_i}\right)\le\frac{\sum_{i=1}^np_if(2a-x_i)}{\sum_{i=1}^np_i}-f\left(\frac{\sum_{i=1}^np_i(2a-x_i)}{\sum_{i=1}^np_i}\right).$$

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


 * $$p_i=1,\ a=\frac{1}{2}, \text{ and } f(x) = \log x. $$