  Let $\K$ be the complete elliptic integral of the first kind. In this paper,
the authors prove that the function $r\mapsto
r^{-2}\{[\log(2\K(r)/\pi)]/\log((\arth r)/r)-3/4\}$ is strictly increasing from
$(0,1)$ onto $(1/320,1/4)$, so that $[(\arth
r)/r]^{3/4+r^2/320}<2\K(r)/\pi<[(\arth r)/r]^{3/4+r^2/4}$ for $r\in(0,1)$, in
which all the coefficients of the exponents of the two bounds are best
possible, thus proving a conjecture raised by Alzer and Qiu to be true, and
giving better bounds of $\K(r)$ than those they conjectured and put in an open
problem. Some other analytic properties of the complete elliptic integrals,
including other kind of approximates for $\K(r)$, are obtained, too.
