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arXiv · 2607.24123

Optimal bounds for the ratio of differences of quadratic, arithmetic, and harmonic means

Zamina E. Guliyeva·Narmin N. Aliyeva·Yagub N. Aliyev
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Abstract

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.

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Zamina E. Guliyeva, Narmin N. Aliyeva, Yagub N. Aliyev. 2026-07-27. Optimal bounds for the ratio of differences of quadratic, arithmetic, and harmonic means. https://arxiv.org/abs/2607.24123

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