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 $\mathcal{E}(r)$, and improve the well-known conjecture $\mathcal{E}(r)>\pi [(1+(1-r^2{)}^{3\mathrm{/}4})\mathrm{/}2{]}^{2\mathrm{/}3}\mathrm{/}2$ for all $r\in (0,1)$ proposed by M. Vuorinen.