Optimal inequalities for a Toader-type mean by quadratic and contraharmonic means

In this paper, we present the best possible parameters α i , β i (i = 1, 2, 3) and α4 , β4 ∈ (1/2, 1) such that the double inequalities { Mathematical Formulas } hold for all a, b 0 with a ≠ b, where Q(a, b), C(a, b), and T(a, b) are the quadratic, contraharmonic, and Toader means, respectively, and TQ,C(a, b) = T[Q(a, b), C(a, b)]. As consequences, we provide new bounds for the complete elliptic integral of econd kind.