On Alzer and Qiu's Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function
On Alzer and Qiu's Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function
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DOI：
10.1155/2011/697547
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发表时间：
2011-01-01
影响因子：
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通讯作者：
Qiu, Ye-Fang
中科院分区：
文献类型：
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作者：
Chu, Yu-Ming;Wang, Miao-Kun;Qiu, Ye-Fang
本文证明了二重不等式（pi/2）（arthr/r）（3/4+alpha*r）< K（r）<（pi/2）（arthr/r）（3/4+beta*r）对所有r是（0，1）中的元素且最佳常数alpha* = 0和beta* = 1/4成立，从而回答了Alzer和Qiu提出的一个公开问题.这里，K（r）是第一类完全椭圆积分，阿尔斯是反双曲正切函数。
We prove that the double inequality (pi/2) (arthr/r)(3/4+alpha*r) < K(r) < (pi/2) (arthr/r)(3/4+beta*r) holds for all r is an element of (0, 1) with the best possible constants alpha* = 0 and beta* = 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.