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

In this paper, for every unital ∗-ring S, we investigate the Jensen‘s inequality preserving maps on C*-convex subsets of S, which we call them C*-convex maps on 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 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.