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    • Keywords


      Root-square mean; arithmetic mean; Toader mean; complete elliptic integrals.

    • Abstract


      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.

    • Author Affiliations


      Yu-Ming Chu1 Miao-Kun Wang2 Song-Liang Qiu3

      1. Department of Mathematics and Computing Science, Hunan City University, Yiyang 413000, China
      2. Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
      3. Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China
    • Dates

  • Proceedings – Mathematical Sciences | News

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      Posted on July 25, 2019

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