We present various inequalities for the complete elliptic integral of the first kind, \[ K ( r ) = ∫ 0 π / 2 1 1 − r 2 sin 2 ⁡ ( t ) d t ( 0 > r > 1 ) . \mathcal {K}(r)=\int _0^{\pi /2} \frac {1}{\sqrt {1- r ^2\sin ^2(t)}}dt \quad {(0>r>1)}. \] Among others, we prove that the inequalities \[ 1 1 + 1 4 r > K ( r ) K ( r ) and K ( 1 − r 2 ) K ( 1 − r ) > 2 1 + r \frac {1}{1+\frac {1}{4}r}>\frac {\mathcal {K}(r)}{\mathcal {K}(\sqrt {r})} \quad \mbox {and} \quad { \frac {\mathcal {K}(\sqrt {1-r^2})}{\mathcal {K}(\sqrt {1-r})}>\frac {2}{1+\sqrt {r}}} \] are valid for all r ∈ ( 0 , 1 ) r\in (0,1) . These estimates refine results published by Anderson, Vamanamurthy, and Vuorinen in 1990.