Enhanced inequalities about arithmetic and geometric means

2020/07/21 by Dai, Fang, Xia, Li-Gang
#FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2008.04067

Abstract

For n positive numbers (a_k, 1≤ k ≤ n), enhanced inequalities about the arithmetic mean (A_n ≡ (∑_ka_k)/(n)) and the geometric mean (G_n≡ √[n]Π_ka_k) are found if some numbers are known, namely, (G_n)/(A_n) ≤ (\fracn-∑_k=1^mr_kn-m)^1-(m)/(n)(Π_k=1^mr_k)^(1)/(n) , if we know a_k=A_nr_k (1≤ k≤ m≤ n) for instance, and (G_n)/(A_n) ≤ \frac1(1-(m)/(n))Π_k=1^mr_k^(-1)/(n-m)+(1)/(n)∑_k=1^mr_k ,if we know a_k=G_nr_k (1≤ k≤ m ≤ n) for instance. These bounds are better than those derived from S.~H.~Tung's work [1].

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