论文标题
关于算术和几何方式的不平等
Enhanced inequalities about arithmetic and geometric means
论文作者
论文摘要
对于$ n $阳性数字（$ a_k $，$ 1 \ leq k \ leq n $），对算术平均值（$ a_n \ equiv \ equiv \ equaC {\ sum_ka_k} {n} {n} $）的不平等增强即，\ begin {equation} \ frac {g_n} {a_n} \ leq （\ frac {n- \ sum_ {k = 1}^mr_k} {n-m}）^{1- \ frac {m} {n} {n}}}（π_{k = 1}^mr_k） $ a_k = a_nr_k $（$ 1 \ leq k \ leq m \ leq n $） \ frac {1} {（1- \ frac {m} {n}）π_{k = 1}^mr_k^{\ frac {-1} {-1} {n-m}}+\ \\ \\ \\ frac {1} {n} {n} {n} {n} {k = 1} $ a_k = g_nr_k $（$ 1 \ leq k \ leq m \ leq n $）。这些界限要比S.〜H.〜Tung的工作[1]更好。
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 ($G_n\equiv \sqrt[n]{Π_ka_k}$) 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}}(Π_{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})Π_{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].
