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

2026/07/27 by Zamina E. Guliyeva, Narmin N. Aliyeva, Yagub N. Aliyev
Mathematics · #math.CA

paper · pdf

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≥3 the sharp double inequality (1)/(√(n))≤ (A_n-H_n)/(Q_n-H_n)≤ √((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.

Citations

• The extremal values of the ratio of differences of power mean, arithmetic mean, and geometric mean

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