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)E(r), and improve the well-known conjecture E(r)>π[(1+(1−r2)3/4)/2]2/3/2E(r)>π[(1+(1−r2)3/4)/2]2/3/2 for all r∈(0,1)r∈(0,1) proposed by M. Vuorinen.