BOUNDS FOR THE GENERALIZED ELLIPTIC INTEGRAL OF THE SECOND KIND.

For a ∈ (0,1) and r ∈ (0,1), let Ea(r) be the generalized elliptic integral of the second kind and R(a) be Ramanujan's constant. In this paper, we prove the following inequalities sin(πa) 2(1-a) +r'2 (1-a)sin(πa)log e R(a)/2 r ! -σ ! < Ea(r) < sin(πa) 2(1-a) +r 2 (1-a)sin(πa)log e R(a)/2 r ! -δ ! with the best possible constants σ = 1 4 sin(πa) and δ = sin(πa) 2(1-a) + (1-a)sin(πa) R(a) 2 - π 2. [ABSTRACT FROM AUTHOR]