Kantorovich Inequality

Suppose $x_1<x_2<\ldots<x_n$ are given Positive numbers. Let $\lambda_1$ , ..., $\lambda_n\geq 0$ and $\sum_{j=1}^n \lambda_j =1$ . Then

$\begin{displaymath} \left({\sum_{j=1}^n \lambda_j x_j}\right)\left({\sum_{j=1}^n\lambda_j{x_j}^{-1}}\right)\leq A^2G^{-2}, \end{displaymath}$

where

$\displaystyle A$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}(x_1+x_n)$
$\displaystyle G$	$\textstyle =$	$\displaystyle \sqrt{x_1x_n}$

are the Arithmetic and Geometric Mean, respectively, of the first and last numbers.

References

Pták, V. ``The Kantorovich Inequality.'' Amer. Math. Monthly 102, 820-821, 1995.