The Generalized Inequalities via Means and Positive Linear Mappings

In this paper, we establish further improvements of the Young inequality and its reverse. Then, we assert operator versions corresponding them. Moreover, an application including positive linear mappings is given. For example, if A, B is an element of B(H) are two invertible positive operators such that 0 < m <= A <= m' < M ' <= B <= M or 0 < m <= B <= m' < M ' <= A <= M for some positive real numbers m, m', M, M ' and let F be a positive unital linear map, then for every 0 <= nu < mu <= 1 [GRAPHICS] where r = min{nu, 1 - nu}, K(h) = (1+h)(2)/4h, h = M/m, h' = M'/m' and r' = min{2r, 1- 2r}. The results of this paper generalize the results of recent years.