Optimal Convex Combination Bounds for Toader Mean
Shao-yun Li
Teachers Teaching Development Center, Wenzhou Broadcast and TV University, Wenzhou 325000, China.
Hui-zuo Xu *
Teachers Teaching Development Center, Wenzhou Broadcast and TV University, Wenzhou 325000, China and Lifelong Education Guidance Center, Wenzhou Broadcast and TV University, Wenzhou 325000, China.
Fang Jin
Teachers Teaching Development Center, Wenzhou Broadcast and TV University, Wenzhou 325000, China and Lifelong Education Guidance Center, Wenzhou Broadcast and TV University, Wenzhou 325000, China.
*Author to whom correspondence should be addressed.
Abstract
In this paper, the authors prove that the double inequalities
hold for all a, b > 0 with 
if and only if α1 ≤ 1/2, β1 ≥ 3/5,α2 ≤ 1/3 and β2 ≥1/2.Here T(a, b),T [A(a, b), G(a, b)],H(a, b) and G(a, b) are the Toader, Toader-type, harmonic and geometric means of a and b , respectively.
Keywords: Toader mean, toader-type mean, harmonic mean, geometric mean, the complete elliptic integral