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The best bounds for Toader mean in terms of the centroidal and arithmetic means

Hua Yun (Department of Information Engineering, Weihai Vocational College, Weihai City, Shandong Province, China)
Qi Feng (College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China + Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, China)

In the paper, the authors discover the best constants α1, α2, β1, and β2 for the double inequalities α1C(a,b) + (1-α1)A(a,b) < T(a,b) < β1C(a,b) + (1-β1)A(a,b) and α2/A(a,b) + 1-α2/C(a,b) < 1/T(a,b) < β2/A(a,b) + 1-β2-C(a,b) to be valid for all a, b > 0 with a ≠ b, where C(a,b) = 2(a2+ab+b2)/3(a+b), A(a,b) = a+b/2, and T(a,b) = 2/π∫π2,0 √a2 cos2 θ + b2 sin2 θ d θ are respectively the centroidal, arithmetic, and Toader means of two positive numbers a and b. As an application of the above inequalities, the authors also find some new bounds for the complete elliptic integral of the second kind.

Keywords: Toader mean, complete elliptic integrals, arithmetic mean, centroidal mean