A sharp lower bound for the complete elliptic integrals of the first kind
被引:9
作者:
Yang, Zhen-Hang
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机构:
North China Elect Power Univ, Engn Res Ctr Intelligent Comp Complex Energy Syst, Minist Educ, Yonghua St 619, Baoding 071003, Peoples R China
Zhejiang Elect Power Co, Res Inst, Hangzhou 310014, Peoples R ChinaNorth China Elect Power Univ, Engn Res Ctr Intelligent Comp Complex Energy Syst, Minist Educ, Yonghua St 619, Baoding 071003, Peoples R China
Yang, Zhen-Hang
[1
,2
]
Tian, Jing-Feng
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h-index: 0
机构:
North China Elect Power Univ, Dept Math & Phys, Yonghua St 619, Baoding 071003, Peoples R ChinaNorth China Elect Power Univ, Engn Res Ctr Intelligent Comp Complex Energy Syst, Minist Educ, Yonghua St 619, Baoding 071003, Peoples R China
Tian, Jing-Feng
[3
]
Zhu, Ya-Ru
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h-index: 0
机构:
North China Elect Power Univ, Dept Math & Phys, Yonghua St 619, Baoding 071003, Peoples R ChinaNorth China Elect Power Univ, Engn Res Ctr Intelligent Comp Complex Energy Syst, Minist Educ, Yonghua St 619, Baoding 071003, Peoples R China
Zhu, Ya-Ru
[3
]
机构:
[1] North China Elect Power Univ, Engn Res Ctr Intelligent Comp Complex Energy Syst, Minist Educ, Yonghua St 619, Baoding 071003, Peoples R China
[2] Zhejiang Elect Power Co, Res Inst, Hangzhou 310014, Peoples R China
[3] North China Elect Power Univ, Dept Math & Phys, Yonghua St 619, Baoding 071003, Peoples R China
Arithmetic-geometric mean;
Logarithmic mean;
Complete elliptic integrals of the first kind;
Inverse hyperbolic tangent function;
NP type power series;
Inequality;
FUNCTIONAL INEQUALITIES;
MONOTONICITY;
CONVEXITY;
D O I:
10.1007/s13398-020-00949-6
中图分类号:
O1 [数学];
学科分类号:
0701 ;
070101 ;
摘要:
Let K(r) be the complete elliptic integrals of the first kind and arthr denote the inverse hyperbolic tangent function. We prove that the inequality 2/pi K(r) > [1 - lambda +lambda (arthr/r)(q)](1/q) holds for r is an element of (0, 1) with the best constants lambda = 3/4 and q = 1/10. This improves some known results and gives a positive answer for a conjecture on the best upper bound for the Gaussian arithmetic-geometric mean in terms of logarithmic and arithmetic means.