关键词: Complete elliptic integrals of the first kind, absolute monotonicity, hypergeometric series, recurrence method, inequality

Abstract: Let $\mathcal{K}\left( r\right) $ be the complete elliptic integrals of the first kind for $r\in \left( 0,1\right) $ and $f_{p}\left( x\right) =\left[ \left( 1-x\right) ^{p}\mathcal{K}\left( \sqrt{x}\right) \right] $. Using the recurrence method, we find the necessary and sufficient conditions for the functions $-f_{p}^{\prime }$, $\ln f_{p}$, $-\left( \ln f_{p}\right) ^{\left( i\right) }$ ($i=1,2,3$) to be absolutely monotonic on $\left( 0,1\right) $. As applications, we establish some new bounds for the ratios and the product of two complete integrals of the first kind, including the double inequalities \begin{eqnarray*} \frac{\exp \left[ r^{2}\left( 1-r^{2}\right) /64\right] }{\left( 1+r\right) ^{1/4}} &<&\frac{\mathcal{K}\left( r\right) }{\mathcal{K}\left( \sqrt{r}% \right) }<\exp \left[ -\frac{r\left( 1-r\right) }{4}\right] , \\ \frac{\pi }{2}\exp \left[ \theta _{0}\left( 1-2r^{2}\right) \right] &<&\frac{% \pi }{2}\frac{\mathcal{K}\left( r^{\prime }\right) }{\mathcal{K}\left( r\right) }<\frac{\pi }{2}\left( \frac{r^{\prime }}{r}\right) ^{p}\exp \left[ \theta _{p}\left( 1-2r^{2}\right) \right] , \\ \mathcal{K}^{2}\left( \frac{1}{\sqrt{2}}\right) &\leq &\mathcal{K}\left( r\right) \mathcal{K}\left( r^{\prime }\right) \leq \frac{1}{\sqrt{% 2rr^{\prime }}}\mathcal{K}^{2}\left( \frac{1}{\sqrt{2}}\right) \end{eqnarray*}% for $r\in \left( 0,1\right) $ and $p\geq 13/32$, where $r^{\prime }=% \sqrt{1-r^{2}}$ and $\theta _{p}=2\Gamma \left( 3/4\right) ^{4}/\pi ^{2}-p$.

Key words: Complete elliptic integrals of the first kind, absolute monotonicity, hypergeometric series, recurrence method, inequality