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

Abstract

In this paper, we present the best possible parameters α i , β i (i = 1, 2, 3) and α 4 , β 4 ∈ (1/2, 1) such that the double inequalitieshold 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, andAs consequences, we provide new bounds for the complete elliptic integral of the second kind.