More about operator order preserving

It is well known that increasing functions do not preserve operator order in general; nor decreasing reverse order. However, monotone or do. In this article, we employ a convex approach to discuss preserving conversing functions. As an easy consequence of more general results, find nonnegative constants γ and ψ such A≤B implies f(B)≤f(A)+γ1ℋ andf(A)≤f(B)+ψ1ℋ, for the self-adjoint operators A, B on Hilbert space ℋ with identity 1ℋ function f whose domain contains spectra both A B. The connection these results existing literature will be discussed significance emphasized by some examples.