  In this paper, we establish several asymptotical bounds for the complete
elliptic integrals of the second kind $\mathcal{E}(r)$, and improve the
well-known conjecture $\mathcal{E}(r)>\pi[(1+(1-r^2)^{3/4})/2]^{2/3}/2$ for all
$r\in(0,1)$ proposed by M. Vuorinen.
