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Sharp Generalized Seiffert Mean Bounds for Toader Mean

Author

Listed:
  • Yu-Ming Chu
  • Miao-Kun Wang
  • Song-Liang Qiu
  • Ye-Fang Qiu

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 ‘ .

Suggested Citation

  • Yu-Ming Chu & Miao-Kun Wang & Song-Liang Qiu & Ye-Fang Qiu, 2011. "Sharp Generalized Seiffert Mean Bounds for Toader Mean," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-8, October.
    • Handle: RePEc:hin:jnlaaa:605259
    DOI: 10.1155/2011/605259
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    Cited by:

    1. Yang, Zhen-Hang & Chu, Yu-Ming & Zhang, Wen, 2019. "High accuracy asymptotic bounds for the complete elliptic integral of the second kind," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 552-564.

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