On a Conjecture Concerning the Approximates of Complete Elliptic Integral of the First Kind by Inverse Hyperbolic Tangent

Let $\text{K}$ be the complete elliptic integral of the first kind. In this paper, the authors prove that the function $r\mapsto r^{-2}{[\log(2\K®/\pi)]/\log((\arth r)/r)-3/4}$isstrictlyincreasingfrom$(0,1)$onto$(1/320,1/4)$,sothat$[(\arth r)/r]^{3/4+r^2/320}<2\K®/\pi<[(\arth r)/r]^{3/4+r^2/4}$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 $\text{K}(r)$ 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 $\text{K}(r)$, are obtained, too.