Enhanced inequalities about arithmetic and geometric means

TL;DR

This paper introduces improved inequalities relating the arithmetic and geometric means of positive numbers, providing tighter bounds when some values are known, surpassing previous results by Tung.

Contribution

The paper derives new, sharper inequalities connecting arithmetic and geometric means using partial information, advancing the theoretical understanding of mean inequalities.

Findings

Derived bounds are tighter than previous inequalities.

Applicable when some numbers are known as either arithmetic or geometric means.

Enhances the theoretical framework for inequalities involving means.

Abstract

For $n$ positive numbers ($a_k$, $1\leq k \leq n$), enhanced inequalities about the arithmetic mean ($A_n \equiv \frac{\sum_ka_k}{n}$) and the geometric mean ($G_n\equiv \sqrt[n]{\Pi_ka_k}$) are found if some numbers are known, namely, \begin{equation} \frac{G_n}{A_n} \leq (\frac{n-\sum_{k=1}^mr_k}{n-m})^{1-\frac{m}{n}}(\Pi_{k=1}^mr_k)^{\frac{1}{n}} \:, \nonumber \end{equation} if we know $a_k=A_nr_k$ ($1\leq k\leq m\leq n$) for instance, and \begin{equation} \frac{G_n}{A_n} \leq \frac{1}{(1-\frac{m}{n})\Pi_{k=1}^mr_k^{\frac{-1}{n-m}}+\frac{1}{n}\sum_{k=1}^mr_k} \: ,\nonumber \end{equation} if we know $a_k=G_nr_k$ ($1\leq k\leq m \leq n$) for instance. These bounds are better than those derived from S.~H.~Tung's work [1].