On Alzer and Qiu’s conjecture for complete elliptic integral and inverse hyperbolic tangent function. (English) Zbl 1226.33010
Summary: We prove that the double inequality
\[ (\pi/2)(\operatorname{arth}r/r)^{3/4+\alpha^*r} <{\mathcal K}(r)< (\pi/2)(\operatorname{arth}r/r)^{3/4+\beta^*r} \]
holds for all \(r\in(0, 1)\) with the best possible constants \(\alpha^* = 0\) and \(\beta^* = 1/4\), which answers to an open problem proposed by Alzer and Qiu. Here, \({\mathcal K}(r)\) is the complete elliptic integral of the first kind, and arth is the inverse hyperbolic tangent function.
\[ (\pi/2)(\operatorname{arth}r/r)^{3/4+\alpha^*r} <{\mathcal K}(r)< (\pi/2)(\operatorname{arth}r/r)^{3/4+\beta^*r} \]
holds for all \(r\in(0, 1)\) with the best possible constants \(\alpha^* = 0\) and \(\beta^* = 1/4\), which answers to an open problem proposed by Alzer and Qiu. Here, \({\mathcal K}(r)\) is the complete elliptic integral of the first kind, and arth is the inverse hyperbolic tangent function.
MSC:
| 33E05 | Elliptic functions and integrals |
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