• Song-Liang Qiu

      Articles written in Proceedings – Mathematical Sciences

    • Optimal Combinations Bounds of Root-Square and Arithmetic Means for Toader Mean

      Yu-Ming Chu Miao-Kun Wang Song-Liang Qiu

      More Details Abstract Fulltext PDF

      We find the greatest values $\alpha_1$ and $\alpha_2$, and the least values $\beta_1$ and $\beta_2$, such that the double inequalities $\alpha_1S(a,b)+(1-\alpha_1)A(a,b) < T(a,b) < \beta_1S(a,b)+(1-\beta_1)A(a,b)$ and $S^{\alpha_2}(a,b)A^{1-\alpha_2}(a,b) < T(a,b) < S^{\beta_2}(a,b)A^{1-\beta_2}(a,b)$ hold for all $a,b>0$ with $a\neq b$. As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions. Here, $S(a,b)=[(a^2+b^2)/2]^{1/2},A(a,b)=(a+b)/2$, and $T(a,b)=\frac{2}{\pi}\int^{\pi/2}_{0}\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}d\theta$ denote the root-square, arithmetic, and Toader means of two positive numbers 𝑎 and 𝑏, respectively.

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