On Alzer and Qiu's Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function

Yu-Ming Chu, Miao-Kun Wang and Ye-Fang Qiu

Abstract and Applied Analysis, 2011, vol. 2011, 1-7

Abstract:

We prove that the double inequality ( 𠜋 / 2 ) ( a r t h 𠑟 / 𠑟 ) 3 / 4 + 𠛼 ∗ 𠑟 < 𠒦 ( 𠑟 ) < ( 𠜋 / 2 ) ( a r t h 𠑟 / 𠑟 ) 3 / 4 + 𠛽 ∗ 𠑟 holds for all 𠑟 ∈ ( 0 , 1 ) with the best possible constants 𠛼 ∗ = 0 and 𠛽 ∗ = 1 / 4 , which answer to an open problem proposed by Alzer and Qiu. Here, 𠒦 ( 𠑟 ) is the complete elliptic integrals of the first kind, and arth is the inverse hyperbolic tangent function.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:697547

DOI: 10.1155/2011/697547

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