Operator inequalities: From a general theorem to concrete inequalities

Abstract The aim of this paper is to give a method to extract concrete inequalities from a general theorem, which is established by making use of majorization relation between functions. By this method we can get a lot of inequalities; among others we extend Furuta inequality as follows: Let f i , g j be positive operator monotone functions on [ 0 , ∞ ) and put k ( t ) = t r 0 f 1 ( t ) r 1 ⋯ f m ( t ) r m , h ( t ) = t p 0 g 1 ( t ) p 1 ⋯ g n ( t ) p n , where p 0 ≥ 1 and r i ≥ 0 , p j ≥ 0 . Then 0 ≤ A ≤ C ≤ B implies, for 0 α ≤ 1 + r 0 p + r 0 with p = p 0 + ⋯ + p n , ( k ( C ) 1 2 h ( A ) k ( C ) 1 2 ) α ≤ ( k ( C ) 1 2 h ( C ) k ( C ) 1 2 ) α ≤ ( k ( C ) 1 2 h ( B ) k ( C ) 1 2 ) α . Moreover, we show log ⁡ C 1 / 2 e A C 1 / 2 ≤ log ⁡ C 1 / 2 e C C 1 / 2 ≤ log ⁡ C 1 / 2 e B C 1 / 2 , provided C is invertible. We also refer to operator geometric mean.