Sharp Approximations for Complete p-Elliptic Integral of the Second Kind by Weighted Power Means

In this paper, the well-known double inequality for the complete elliptic integral E ( r ) of the second kind, which gives sharp approximations of E ( r ) by power means (or Hölder means), is extended to the complete p -elliptic integral E p ( r ) of the second kind, and thus sharp approximations of E p ( r ) by weighted power means are obtained. This result confirmed the truth of Conjecture I by Barnard, Ricards and Tiedeman in the case when a = b = 1 / p ∈ ( 0 , 1 / 2 ) and c = 1 and also provides a new method to prove the above double inequality of E ( r ).