Nonlinear mixed skew and \(\eta \)-Jordan derivations on \(*\)-algebras

Let \(\mathfrak {E}\) be a unital \(*\)-algebra and \(\eta \ne -1\) be a nonzero scalar. This paper establishes that if a nonlinear mapping \(\zeta : \mathfrak {E} \rightarrow \mathfrak {E}\) satisfies \( \zeta (\mathcal {U} \bullet \mathcal {V} \circ _\eta \mathcal {W})=\zeta (\mathcal {U}) \bullet \mathcal {V} \circ _\eta \mathcal {W}+\mathcal {U} \bullet \zeta (\mathcal {V}) \circ _\eta \mathcal {W}+\mathcal {U} \bullet \mathcal {V} \circ _\eta \zeta (\mathcal {W})\) for all \( \mathcal {U}, \mathcal {V}, \mathcal {W} \in \mathfrak {E}\), then \(\zeta \) is an additive \(*\)-derivation and fulfils \(\zeta (\eta \mathcal {U})=\eta \zeta (\mathcal {U})\) for all \(\mathcal {U} \in \mathfrak {E}.\) Additionally, we extend this result to various other algebras.