当柯西不等式成为公式时
When the Cauchy inequality becomes a formula
摘要：
本文重新讨论了经典的几何-算术平均不等式，并给出了给定$n\in\mathbb N$非负数$x_1,x_2,\dots,x_n$的算术平均数与几何平均数之差的一个公式.该公式产生了新的更强版本的几何-算术平均不等式。我们还发现了一个强几何-算术平均不等式的第二个版本，并证明了所有不等式在某种意义下都是最优的。另一个显著的新颖性是，所有新的不等式中的相等性不仅在所有$n$数相等的情况下成立，而且在其他情况下也成立。
英文摘要：
In this note we revisit the classical geometric-arithmetic mean inequality and find a formula for the difference of the arithmetic and the geometric means of given $n\in\mathbb N$ nonnegative numbers $x_1,x_2,\dots,x_n$. The formula yields new stronger versions of the geometric-arithmetic mean inequality. We also find a second version of a strong geometric-arithmetic mean inequality and show that all inequalities are optimal in some sense. Anther striking novelty is, that the equality in all new inequalities holds not only in the case when all $n$ numbers are equal, but also in other cases.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
