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We prove that the double inequality (π/2)((π/2)( holds for all r ∈ (0, 1) with the best possible constants α* = 0 and β* = 1/4, which answer to an open problem proposed by Alzer and Qiu. Here, 𝒦(r) is the complete elliptic integrals of the first kind, and arth is the inverse hyperbolic tangent function.
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Chu et al. (2011) studied this question.
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