An Operator Extension of Čebyšev Inequality

Some operator inequalities for synchronous functions that are related to the čebyšev inequality are given. Among other inequalities for synchronous functions it is shown that ∥ø(f(A)g(A)) - ø(f(A))ø(g(A))∥ ≤ max{║ø(f²(A)) - ø²(f(A))║, ║ø)G²(A)) - ø²(g(A))║} where A is a self-adjoint and compact operator on B(ℋ ), f, g ∈ C (sp (A)) continuous and non-negative functions and ø: B(ℋ ) → B(ℋ ) be a n-normalized bounded positive linear map. In addition, by using the concept of quadruple D-synchronous functions which is generalizes the concept of a pair of synchronous functions, we establish an inequality similar to čebyšev inequality.