Optimal bounds for a Gaussian Arithmetic-Geometric 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 α_4,β_4∈(1/2,1) such that the double inequalities α_1Q(a,b)+(1-α_1)C(a,b) 0 with a≠ b, where Q(a,b), C(a,b) and AG(a,b) are the quadratic, contraharmonic and Arithmetic-Geometric means, and AG_Q,C(a,b)=AG[Q(a,b),C(a,b)]. As consequences, we present new bounds for the complete elliptic integral of the first kind. Keywords: Arithmetic-Geometric mean, Complete elliptic integral, Quadratic mean, Contraharmonic mean