Sharp bounds for the complete elliptic integral of the first kind in term of the inverse tangent hyperbolic function

Let \(\mathcal{K}\)(r) and arctanh r for r ∈ (0, 1) be the complete elliptic integral of the first kind and the inverse tangent hyperbolic function, respectively. In this paper, we prove that the double inequality

\({\Phi }_{p}\left({r}{\prime}\right)\frac{\text{arctanh}r}{r}<\frac{2}{\pi }\mathcal{K}\left(r\right)<{\Phi }_{q}\left({r}{\prime}\right)\frac{\text{arctanh}r}{r}\)

holds for r ∈ (0, 1) if and only if q ⩽ 56 543/20 976 and 23(90π − 233)/(10(69π − 178)) ⩽ p ⩽ 3, where r′ \(\sqrt{1-{r}^{2}}\) and

\({\Phi }_{q}\left(x\right)=60\frac{\left(17q-41\right){x}^{2}+6qx+69-23q}{\left(620q-1521\right){x}^{2}+2\left(580q-1079\right)x+5359-1780q}\)

for q ⩽ 3 and x ∈ (0, 1). This improves some known results and yields several new bounds for the Gauss arithmetic–geometric mean.