Further generalizations of some operator inequalities involving positive linear map

Abstract

We obtain a generalized conclusion based on an ?-geometric mean inequality. The conclusion is presented as follows: If m1,M1,m2,M2 are positive real numbers, 0 < m1 ? A ? M1 and 0 < m2 ? B ? M2 for m1 < M1 and m2 < M2, then for every unital positive linear map ? and ? ? (0,1], the operator inequality below holds: (?(?)#??(B))p ? 1/16 {(M1+m1)2((M1+m1)-1(M2+m2))2?)/(m2M2)?(m1M1)1- ?}p ?p(A#?B), p ? 2. Likewise, we give a second powering of the Diaz-Metcalf type inequality. Finally, we present p-th powering of some reversed inequalities for n operators related to Karcher mean and power mean involving positive linear maps.