An inequality generalizing $AM\ge GM$ and $AM\ge HM$

Abstract

Aim of this article is to prove the inequality $n{\sum }_{i=1}^n{a}_i{b}_i\le {\sum }_{i=1}^n{a}_i{\sum }_{i=1}^n{b}_i$ when $a_i$ are $n$ increasing positive real numbers and $b_i$ are $n$ decreasing real numbers. We also prove generalizations of this result.

Known results