Sharp Generalized Seiffert Mean Bounds for Toader Mean

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

Abstract and Applied Analysis, 2011, vol. 2011, 1-8

Abstract:

For ð ‘ âˆˆ [ 0 , 1 ] , the generalized Seiffert mean of two positive numbers ð ‘Ž and ð ‘ is defined by 𠑆 ð ‘ ( ð ‘Ž , ð ‘ ) = ð ‘ ( ð ‘Ž − ð ‘ ) / a r c t a n [ 2 ð ‘ ( ð ‘Ž − ð ‘ ) / ( ð ‘Ž + ð ‘ ) ] , 0 < ð ‘ â‰¤ 1 , ð ‘Ž â‰ ð ‘ ; ( ð ‘Ž + ð ‘ ) / 2 , ð ‘ = 0 , ð ‘Ž â‰ ð ‘ ; ð ‘Ž , ð ‘Ž = ð ‘ . In this paper, we find the greatest value ð ›¼ and least value ð ›½ such that the double inequality 𠑆 ð ›¼ ( ð ‘Ž , ð ‘ ) < 𠑇 ( ð ‘Ž , ð ‘ ) < 𠑆 ð ›½ ( ð ‘Ž , ð ‘ ) holds for all ð ‘Ž , ð ‘ > 0 with ð ‘Ž â‰ ð ‘ , and give new bounds for the complete elliptic integrals of the second kind. Here, ∫ 𠑇 ( ð ‘Ž , ð ‘ ) = ( 2 / 𠜋 ) 0 𠜋 / 2 √ ð ‘Ž 2 c o s 2 𠜃 + ð ‘ 2 s i n 2 𠜃 ð ‘‘ 𠜃 denotes the Toader mean of two positive numbers ð ‘Ž and ð ‘ .

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:605259

DOI: 10.1155/2011/605259

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