AI 中文总结
研究\(n\)个非负实数二次、算术和调和平均差之比的最优界，采用基于Cauchy和Maclaurin经典优化方法变体，证明\(n\geq3\)时的双重不等式，扩展了T. Mitev仅针对\(n = 3、4、5\)的结果。
AI 中文摘要
我们确定了比较\(n\)个非负实数的二次、算术和调和平均差的不等式中的最优常数。具体而言，我们证明对于\(n\geq3\)，尖锐的双重不等式\(\frac{1}{\sqrt{n}}\leq\frac{A_n - H_n}{Q_n - H_n}\leq\sqrt{\frac{n - 1}{n}}\)成立。这扩展了T. Mitev早期的结果，他仅针对\(n = 3、4\)和\(5\)的情况建立了尖锐界。我们的方法基于Cauchy和Maclaurin经典优化方法的一个变体，其中在算术和调和平均的同时约束下优化二次对称函数。
英文摘要
We determine the optimal constants in inequalities comparing the differences of the quadratic, arithmetic, and harmonic means of $n$ nonnegative real numbers. Specifically, we prove that for $n\ge3$ the sharp double inequality \[ \frac{1}{\sqrt{n}}\le \frac{A_n-H_n}{Q_n-H_n}\le \sqrt{\frac{n-1}{n}} \] holds true. This extends earlier results by T. Mitev, which established the sharp bounds only for the cases $n=3,4,$ and $5$. Our approach is based on a variant of the classical optimization method of Cauchy and Maclaurin, in which a quadratic symmetric function is optimized under simultaneous constraints on the arithmetic and harmonic means.
Comments10 pages, 2 figures
