On Jordan ∗-mappings in rings with involution

The objective of this paper is to study Jordan ∗∗-mappings in rings with involution ∗∗. In particular, we prove that if R   is a prime ring with involution ∗∗, of characteristic different from 2 and D   is a nonzero Jordan ∗∗-derivation of R   such that [D(x),x]=0[D(x),x]=0, for all x∈Rx∈R and S(R)∩Z(R)≠(0), then R   is commutative. Further, we also prove a similar result in the setting of Jordan left ∗∗-derivation. Finally, we prove that any symmetric Jordan triple ∗∗-biderivation on a 2-torsion free semiprime ring with involution ∗∗ is a symmetric Jordan ∗∗-biderivation.