A generalization of Ky Fan's inequality

Peng Gao

International Journal of Mathematics and Mathematical Sciences, 2001, vol. 28, 1-7

Abstract:

Let P n , r ( x ) be the generalized weighted means. Let F ( x ) be a C 1 function, y = y ( x ) an implicit decreasing function defined by f ( x , y ) = 0 and 0 < m < M ≤ m ′ , n ≥ 2 , x i ∈ [ m , M ] , y i ∈ [ m ′ , M ′ ] . Then for − 1 ≤ r ≤ 1 , if f ′ x / f ′ y ≤ 1 , | ( F ( P n , 1 ( y ) ) − F ( P n , r ( y ) ) ) / ( F ( P n , 1 ( x ) ) − F ( P n , r ( x ) ) ) | < ( max m ′ ≤ ξ ≤ M ′ | F ′ ( ξ ) | ) / ( min m ≤ η ≤ M | F ′ ( η ) | ) ⋅ M / m ′ ⋅ M / m ′ A similar result exists for f ′ x / f ′ y ≥ 1 . By specifying f ( x , y ) and F ( x ) , we get various generalizations of Ky Fan's inequality. We also present some results on the comparison of P n , s α ( y ) − P n , r α ( y ) and P n , s α ( x ) − P n , r α ( x ) for s ≥ r , α ∈ ℝ .

Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:615690

DOI: 10.1155/S0161171201012467

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