Sandor-Yang平均关于算术和二次平均的确界

Bounds for Sandor-Yang Means in Terms of Arithmetic and Quadratic Means

研究Sándor-Yang平均R AQ(或R QA)关于算术平均A与二次平均Q的几何组合的序关系,得到两个精确双向不等式:Q^α1(a,b)A^(1-α1)a,b<R QA a,b<Q^β1(a,b)A^(1-β1)(a,b)和Q^α2(a,b)A^(1-α2)(a,b)<R AQ(a,b)<Q^β2(a,b)A^(1-β2)(a,b)对所有a,b>0且a≠b成立的充要条件是α1≤2/3、β1≥0.7111L,α2≤1/3和β2≥0.3807L。

This paper study the order relation of geometric combinations of arithmetic mean A and quadratic mean Q for Sándor-Yang mean R AQ(or R QA),and two optimal double inequalities for Sándor-Yang mean are found:Q^α1(a,b)A^(1-α1)a,b<R QA a,b<Q^β1(a,b)A^(1-β1)(a,b)和Q^α2(a,b)A^(1-α2)(a,b)<R AQ(a,b)<Q^β2(a,b)A^(1-β2)(a,b)The necessary and sufficient conditions for all of a,b>0 and a≠b areα1≤2/3,β1≥0.7111L,α2≤1/3 andβ2≥0.3807L.