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

TL;DR

This paper proves a new inequality involving sums of increasing and decreasing sequences and explores its generalizations, extending classical inequalities like AM-GM and AM-HM.

Contribution

It introduces a novel inequality relating sums of increasing and decreasing sequences and provides generalizations of this inequality.

Findings

Established the inequality $n imes ext{sum of } a_ib_i \

Proved the inequality holds for increasing $a_i$ and decreasing $b_i$ sequences.

Presented generalizations of the main inequality.

Abstract

Aim of this article is to prove the inequality $n \sum_{i=1}^{n} a_ib_i \leq \sum_{i=1}^{n} a_i \sum_{i=1}^n b_i$ when $a_i$ are $n$ increasing positive real numbers and $b_i$ are $n$ decreasing real numbers. We also prove generalizations of this result.