  Let $\mathcal{A}$ be a $C^*$-algebra and $\phi:\cA\to L(H)$ be a positive
unital map. Then, for a convex function $f:I\to \mathbb{R}$ defined on some
open interval and a self-adjoint element $a\in \mathcal{A}$ whose spectrum lies
in $I$, we obtain a Jensen's-type inequality $f(\phi(a)) \leq \phi(f(a))$ where
$\le$ denotes an operator preorder (usual order, spectral preorder,
majorization) and depends on the class of convex functions considered i.e.,
operator convex, monotone convex and arbitrary convex functions. Some
extensions of Jensen's-type inequalities to the multi-variable case are
considered.
