A new generalized refinement of the weighted arithmetic-geometric mean inequality

In this paper, we prove that for i = 1,2, . . . ,n , ai 0 and αi > 0 satisfy ∑i=1 αi = 1 , then for m = 1,2,3, . . . , we have ( n ∏ i=1 ai i )m + rm 0 ( n ∑ i=1 ai −n n √ n ∏ i=1 ai ) ( n ∑ i=1 αiai )m where r0 = min{αi : i = 1, . . . ,n} . This is a considerable generalization of the two refinements of the arithmetic-geometric mean inequality due to Furuichi [2], Manasrah and Kittaneh [7], which correspond to the cases m = 1 and n = 2 , respectively. As application we give some generalized inequalities of determinants for positive definite matrices. Mathematics subject classification (2010): 26D07, 26D15, 15A45.