Epigraph of Operator Functions

It is known that a real function $f$ is convex if and only if the set $$\mathrm{E}(f)=\{(x,y)\in \mathbb{R}\times \mathbb{R};f(x)\le y\},$$ the epigraph of $f$ is a convex set in ${\mathbb{R}}^2$. We state an extension of this result for operator convex functions and $C^{\ast }$-convex sets as well as operator log-convex functions and $C^{\ast }$-log-convex sets. Moreover, the $C^{\ast }$-convex hull of a Hermitian matrix has been represented in terms of its eigenvalues.