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

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/122/01/0041-0051

• # 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 $\alpha_1S(a,b)+(1-\alpha_1)A(a,b) &lt; T(a,b) &lt; \beta_1S(a,b)+(1-\beta_1)A(a,b)$ and $S^{\alpha_2}(a,b)A^{1-\alpha_2}(a,b) &lt; T(a,b) &lt; S^{\beta_2}(a,b)A^{1-\beta_2}(a,b)$ hold for all $a,b&gt;0$ 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

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

• # Proceedings – Mathematical Sciences

Volume 133, 2023
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019