Abstract In the article, we present the best possible parameters α1, β1, α2, β2 ∈ ℝ and α3, β3 ∈ [1/2, 1] such that the double inequalities α1C(a,b)+(1−α1)A(a,b) 0 with a ≠ b, and provide new bounds for the complete elliptic integral of the second kind, where A(a, b) = (a + b)/2 is the arithmetic mean, Q(a,b)=a2+b2/2 $\begin{array}{} \displaystyle Q(a, b)=\sqrt{\left(a^{2}+b^{2}\right)/2} \end{array}$ is the quadratic mean, C(a, b) = (a2 + b2)/(a + b) is the contra-harmonic mean, C(p; a, b) = C[pa + (1 – p)b, pb + (1 – p)a] is the one-parameter contra-harmonic mean and T3(a,b)=(2π∫0π/2a3cos2⁡θ+b3sin2⁡θdθ)2/3 $\begin{array}{} T_{3}(a,b)=\Big(\frac{2}{\pi}\int\limits_{0}^{\pi/2}\sqrt{a^{3}\cos^{2}\theta+b^{3}\sin^{2}\theta}\text{d}\theta\Big)^{2/3} \end{array}$ is the Toader mean of order 3.

中文翻译：

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3 阶 Toader 均值在算术、二次和谐波均值方面的锐界

摘要 在本文中，我们提出了最佳可能参数 α1, β1, α2, β2 ∈ ℝ 和 α3, β3 ∈ [1/2, 1] 使得双重不等式 α1C(a,b)+(1−α1)A (a,b) 0 且 a ≠ b，并为第二类完全椭圆积分提供新的界限，其中 A(a, b) = (a + b)/2 是算术平均值，Q(a,b)=a2+b2 /2 $\begin{array}{} \displaystyle Q(a, b)=\sqrt{\left(a^{2}+b^{2}\right)/2} \end{array}$ 是二次均值，C(a, b) = (a2 + b2)/(a + b) 是反谐波均值，C(p; a, b) = C[pa + (1 – p)b, pb + (1 – p)a] 是单参数反谐波均值，T3(a,b)=(2π∫0π/2a3cos2⁡θ+b3sin2⁡θdθ)2/3 $\begin{array}{} T_{ 3}(a,b)=\Big(\frac{2}{\pi}\int\limits_{0}^{\pi/2}\sqrt{a^{3}\cos^{2}\theta +b^{3}\sin^{2}\theta}\text{d}\theta\Big)^{2/3} \end{array}$ 是 3 阶 Toader 均值。