Proceedings - Mathematical Sciences

, Volume 122, Issue 1, pp 41–51

# Optimal combinations bounds of root-square and arithmetic means for Toader mean

• YU-MING CHU
• MIAO-KUN WANG
• SONG-LIANG QIU
Article

## Abstract

We find the greatest value α 1 and α 2, and the least values β 1 and β 2, such that the double inequalities α 1 S(a,b) + (1 − α 1) A(a,b) < T(a,b) < β 1 S(a,b) + (1 − β 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 ≠ 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 $$T(a,b)=\frac{2}{\pi}\int\limits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}{\rm d}\theta$$ denote the root-square, arithmetic, and Toader means of two positive numbers a and b, respectively.

## Keywords

Root-square mean arithmetic mean Toader mean complete elliptic integrals

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