Author： Alzer H.
Publisher： Academic Press
ISSN： 0022-247X
Source： Journal of Mathematical Analysis and Applications, Vol.204, Iss.1, 1996-11, pp. : 291-297
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Abstract
Let x i∈ (0, 1/2] ( i =1,, n ) be real numbers. If A n and G n (respectively, A ' n and G ' n ) denote the weighted arithmetic and geometric means of x 1 ,, x n (respectively, 1- x 1 ,,1- x n ), then eqalign{{A'_nover G'_n}le 1+2(A'_n-G'_n)&le{1-G'_n+A'_nover 1-A'_n+G'_n}cr &le 1+2(A_n-G_n)le{1-G_n+A_nover 1-A_n+G_n}le{A_nover G_n},&hbox{($*$)}cr} with equality holding if and only if x 1 =···= x n . The inequalities (*) provide refinements of Ky Fan's inequality A ' n / G ' n < A n / G n and its additive analogue A ' n - G ' n < A n - G n .
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