关键词: Schwab-Borchardt 平均, Sá, ndor-Yang 平均, 算术平均, 二次平均, 反调和平均

Abstract: This paper deals with the inequalities involving Sándor-Yang means derived from the Schwab-Borchardt mean using the method of real analysis. The convex com- binations of the arithmetic mean A(a,b) and quadratic Q(a,b) (or contra-harmonic mean C(a,b)) for the Sándor-Yang means S_QA(a,b) and S_QA(a,b) are disscused. The main results obtained are the sharp bounds of the two convex combinations, namely, the best possible parameters α₁, α₂, α₃, α₄, β₁, β₂, β₃, β₄ ∈ (0, 1), such that the double inequalities
α₁Q(a,b) + (1-α₁)A(a,b) < S_QA(a,b) < β₁Q(a,b) + (1-β₁)A(a,b),
α₂Q(a,b) + (1-α₂)A(a,b) < S_QA(a,b) < β₂Q(a,b) + (1-β₂)A(a,b),
α₃C(a,b) + (1-α₃)A(a,b) < S_QA(a,b) < β₃C(a,b) + (1-β₃)A(a,b),
α₄C(a,b) + (1-α₄)A(a,b) < S_QA(a,b) < β₄C(a,b) + (1 -β₄)A(a,b)
hold for all a, b > 0 and a≠b. Here A(a,b), Q(a,b) and C(a,b) denote respectively the classical arithmetic, quadratic, contra-harmonic means of a and b, S_QA(a,b) and S_QA(a,b) are two Sándor-Yang means derived from the Schwab-Borchardt mean.

Key words: Schwab-Borchardt mean, Sá, ndor-Yang mean, arithmetic mean, quadratic mean, contra-harmonic mean