Proceedings - Mathematical Sciences

, Volume 122, Issue 1, pp 41–51

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

Article

DOI: 10.1007/s12044-012-0062-y

CHU, YM., WANG, MK. & QIU, SL. Proc Math Sci (2012) 122: 41. doi:10.1007/s12044-012-0062-y

## Abstract

We find the greatest value α1 and α2, and the least values β1 and β2, such that the double inequalities α1S(a,b) + (1 − α1) A(a,b) < T(a,b) < β1S(a,b) + (1 − β1) 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) = [(a2 + b2)/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 meanarithmetic meanToader meancomplete elliptic integrals