Proceedings - Mathematical Sciences

, Volume 122, Issue 1, pp 41–51

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


DOI: 10.1007/s12044-012-0062-y

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


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.


Root-square mean arithmetic mean Toader mean complete elliptic integrals 

Copyright information

© Indian Academy of Sciences 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Computing ScienceHunan City UniversityYiyangChina
  2. 2.Department of MathematicsHuzhou Teachers CollegeHuzhouChina
  3. 3.Department of MathematicsZhejiang Sci-Tech UniversityHangzhouChina

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