In the article, we prove that a = 3, β = log4/(π/2 - log 4)= 7.51371. ··· ,γ = 1/4 and δ = 1 + log2 - π/2 = 0.122351 ··? are the best possible constants such that the double inequalities hold for all r ∈ (0,1), where r' = √1-r~2, and K(r) = ∫_0~(π/2) dθ/√1-r~2sin~2θ and ε(r) =∫_0~(π/2)√1-r~2sin~2θdθ are the complete elliptic integr...
