Acta mathematica scientia,Series B ›› 2022, Vol. 42 ›› Issue (3): 847-864.

• Articles •

### ABSOLUTE MONOTONICITY INVOLVING THE COMPLETE ELLIPTIC INTEGRALS OF THE FIRST KIND WITH APPLICATIONS

Zhenhang YANG1,2, Jingfeng TIAN3

1. 1. Engineering Research Center of Intelligent Computing for Complex Energy Systems of Ministry of Education, North China Electric Power University, Baoding, 071003, China;
2. Zhejiang Society for Electric Power, Hangzhou, 310014, China;
3. Department of Mathematics and Physics, North China Electric Power University, Baoding, 071003, China
• Received:2020-06-05 Revised:2021-08-25 Published:2022-06-24
• Contact: Jingfeng TIAN,E-mail:tianjf@ncepu.edu.en E-mail:tianjf@ncepu.edu.en

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 $\theta _{p}=2\Gamma \left( 3/4\right) ^{4}/\pi ^{2}-p$.

CLC Number:

• 33E05
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