The structure of the set of all C^*-convex maps in *-rings

In this paper, for every unital *-ring \mathcal{S}, we investigate the Jensen's inequality preserving maps on C^*-convex subsets of \mathcal{S}, which we call them C^*-convex maps on \mathcal{S}. We consider an involution for maps on *-rings, and we show that for every C^*-convex map f on the C^*-convex set B in \mathcal{S}, f^* is also a C^*-convex map on B. We prove that  in the unital commutative *-rings, the set of all C^*-convex maps (C^*-affine maps) on a C^*-convex set B, is also a C^*-convex set. In addition, we prove some results for increasing C^*-convex maps. Moreover, it is proved that the set of all C^*-affine maps on B, is a C^*-face of the set of all C^*-convex maps on B in the unital commutative *-rings. Finally, some examples of C^*-convex maps and C^*-affine maps in *-rings are given.