In this paper, we study a special algebraic inequality containing a parameter, the sum of reciprocals and the product of positive real numbers whose sum is 1. Using a new optimization argument the best values of the parameter are determined. In the case of three numbers the algebraic inequality has some interesting geometric applications involving a generalization of Euler’s inequality about the ratio of radii of circumscribed and inscribed circles of a triangle.

中文翻译：

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涉及单位和的正数的倒数和乘积的不等式的最佳常数

在本文中，我们研究了一种特殊的代数不等式，其中包含参数、倒数之和以及总和为 1 的正实数的乘积。使用新的优化参数确定参数的最佳值。在三个数字的情况下，代数不等式有一些有趣的几何应用，涉及关于三角形外接圆和内切圆半径之比的欧拉不等式的推广。