Enhanced inequalities about arithmetic and geometric 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].