More About Operator Order Preserving

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