In the aritcle, we prove that the double inequalities αA(a, b)+(1−α)G(a, b) < T [A(a, b), G(a, b)] < βA(a, b) + (1 − β)G(a, b) and G[λa + (1 − λ)b, λb + (1 − λ)a] < T [A(a, b), G(a, b)] < G[µa + (1 − µ)b, µb + (1 − µ)a] hold for all a, b > 0 with a = b if and only if α ≤ 1/2, β ≥ 2/π, λ ≤ (1 − 1 − 4/π 2)/2 and µ ≥ 1/2 − √ 2/4 if α, β ∈ R and λ, µ ∈ (0, 1/2), and find new bounds for the complete elliptic integral E(r) = π/2 0 (1 − r 2 sin 2 θ) 1/2 dθ (0 < r < 1) of the second kind, where G(a, b) = √ ab,...
