Asymptotical bounds for complete elliptic integrals of the second kind

In this paper, we establish several asymptotical bounds for the complete elliptic integrals of the second kind E ( r ) , and improve the well-known conjecture E ( r ) > π [ ( 1 + ( 1 − r 2 ) 3 / 4 ) / 2 ] 2 / 3 / 2 for all r ∈ ( 0 , 1 ) proposed by M. Vuorinen.