Sharp double inequality for complete elliptic integral of the first kind

For $r\in (0,1)$, the function $\text{K}(r)={\int }_0^{\pi \mathrm{/}2}(1-r^2{\sin }^{2}t{)}^{-1\mathrm{/}2}dt$ is known as the complete elliptic integral of the first kind. In this paper, we prove the absolute monotonicity of two functions involving $\text{K}(r)$. As a consequence, we improve Alzer and Richards’ result.