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Inequalities for the ratio of complete elliptic integrals


Authors: Horst Alzer and Kendall Richards
Journal: Proc. Amer. Math. Soc. 145 (2017), 1661-1670
MSC (2010): Primary 33C75, 39B62; Secondary 26E60
DOI: https://doi.org/10.1090/proc/13337
Published electronically: October 13, 2016
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Abstract: We present various inequalities for the complete elliptic integral of the first kind,

$\displaystyle \mathcal {K}(r)=\int _0^{\pi /2} \frac {1}{\sqrt {1- r ^2\sin ^2(t)}}dt \quad {(0<r<1)}. $

Among others, we prove that the inequalities

$\displaystyle \frac {1}{1+\frac {1}{4}r}<\frac {\mathcal {K}(r)}{\mathcal {K}(\sqrt {r})}$$\displaystyle \quad \mbox {and} \quad { \frac {\mathcal {K}(\sqrt {1-r^2})}{\mathcal {K}(\sqrt {1-r})}<\frac {2}{1+\sqrt {r}}} $

are valid for all $ r\in (0,1)$. These estimates refine results published by Anderson, Vamanamurthy, and Vuorinen in 1990.

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Additional Information

Horst Alzer
Affiliation: Morsbacher Str. 10, D-51545 Waldbröl, Germany
Email: h.alzer@gmx.de

Kendall Richards
Affiliation: Department of Mathematics and Computer Science, Southwestern University, Georgetown, Texas
Email: richards@southwestern.edu

DOI: https://doi.org/10.1090/proc/13337
Keywords: Complete elliptic integrals, functional inequalities, means
Received by editor(s): April 18, 2016
Received by editor(s) in revised form: June 9, 2016, June 10, 2016, and June 12, 2016
Published electronically: October 13, 2016
Communicated by: Mourad Ismail
Article copyright: © Copyright 2016 American Mathematical Society