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Optimal combinations bounds of root-square and arithmetic means for Toader mean

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成果類型:
期刊論文
作者:
Chu, Yu-Ming;Wang, Miao-Kun;Qiu, Song-Liang
通訊作者:
Chu, Y.-M.(chuyuming@hutc.zj.cn)
作者機構:
[Chu, Yu-Ming] Hunan City Univ, Dept Math & Comp Sci, Yiyang 413000, Peoples R China.
[Wang, Miao-Kun] Huzhou Teachers Coll, Dept Math, Huzhou 313000, Peoples R China.
[Qiu, Song-Liang] Zhejiang Sci Tech Univ, Dept Math, Hangzhou 310018, Zhejiang, Peoples R China.
通訊機構:
Department of Mathematics and Computing Science, Hunan City University, China
語種:
英文
關鍵詞:
Root-square mean;arithmetic mean;Toader mean;complete elliptic integrals
期刊:
Proceedings of the Indian Academy of Sciences: Mathematical Sciences
ISSN:
0253-4142
年:
2012
卷:
122
期:
1
頁碼:
41-51
基金類別:
Natural Science Foundation of ChinaNational Natural Science Foundation of China (NSFC) [11071069]; Natural Science Foundation of Hunan ProvinceNatural Science Foundation of Hunan Province [09JJ6003]; Innovation Team Foundation of the Department of Education of Zhejiang Province [T200924]
機構署名:
本校為第一機構
院系歸屬:
理學院
摘要:
We find the greatest values &alpha;<inf>1</inf> and &alpha;<inf>2</inf>, and the least values &beta;1 and &beta;2, such that the double inequalities &alpha;<inf>1</inf>S(a, b) + (1 - &alpha;<inf>1</inf>)A(a, b) &lt;T (a, b) &lt;&beta;<inf>1</inf>S(a, b) + (1 - &beta;<inf>1</inf>)A(a, b) and S<sup>&alpha;2</sup> (a, b)A<sup>1-&alpha;2</sup> (a, b) &lt;T (a, b) &lt;S<sup>&beta;2</sup> (a, b)A<sup>1-&beta;2</sup> (a, b) hold for all a, b &gt;0 with a &ne;b. As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions. Here, S(a, b) = [(a<sup>2</sup> +b<sup>2</sup>)/2]<sup>1/2</sup>, A(a, b) = (a +b)/2, and T (a, b) = 2/&pi;<sup>&pi;/2</sup>&int;<inf>0</inf> &radic;a<sup>2</sup>cos<sup>2</sup> &theta;+ b<sup>2</sup>sin<sup>2</sup> &theta;d&theta;denote the root-square, arithmetic, and Toader means of two positive numbers a and b, respectively. &copy;Indian Academy of Sciences.

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