Sharp inequalities for Toader mean in terms of other bivariate means

In the paper, the author discovers the best constants & alpha;1, & alpha;2, & alpha;3, & beta;1, & beta;2 and & beta;3 for the double inequalities (a - b)2n+2 n n-ary sumation -1 )2k+2 & alpha;1A < T(a, b)-4 1C-3 2,k)2 (1 (a - b (a - b)2n+2 4A-A < & beta;1A a + b 4((k + 1)!)2 a + b a + b k=1 (a - b )2n+2 n n-ary sumation -1 )2k+2 & alpha;2A < T(a, b)-3 4C-1 2, k)2 (1 (a - b (a - b)2n+2 4A-A < & beta;2A a + b 4((k + 1)!)2 a + b a + b k=1 and (a - b)2n+2 n n-ary sumation -1 )2k+2 & alpha;3A < 4 5T (a, b)+1 2, k)2 (1 (a - b (a - b )2n+2 5H-A-A < & beta;3A a + b 5((k + 1)!)2 a + b a + b k=1 to be valid for all a, b > 0 with a = b and n = 1, 2, & BULL; & BULL; & BULL;, where a2 + b2 C & EQUIV; C(a, b) = H & EQUIV; H(a, b) = 2(a2 + ab + b2) a + b a + b , C & EQUIV; C(a, b) = 3(a + b) , A & EQUIV; A(a, b) = 2 , 2ab 2 & int; & pi;/2 & RADIC; a + b, T(a, b) = a2 cos2 & theta; + b2 sin2 & theta; d & theta; & pi; 0 are respectively the contraharmonic, centroidal, arithmetic, harmonic and Toader means of two positive numbers a and b, (a, n) = a(a + 1)(a + 2)(a + 3) & BULL; & BULL; & BULL; (a + n - 1) denotes the shifted factorial function. As an application of the above inequalities, the author also find a new bounds for the complete elliptic integral of the second kind.