  For $r\in(0,1)$, the function $\K(r)=\int_0^{\pi/2}(1-r^2\sin^2t)^{-1/2}dt$
is known as the complete elliptic integral of the first kind. In this paper, we
prove the absolute monotonicity of two functions involving $\K(r)$. As a
consequence, we improve Alzer and Richards' result.
