  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].
