Jordan Homoderivation Behavior of Generalized Derivations in Prime Rings

Suppose that R is a prime ring with char(R) ≠ 2 and f(ξ₁, . . . , ξ_n) is a noncentral multilinear polynomial over C(= Z(U)), where U is the Utumi quotient ring of R. An additive mapping h : R ⟶ R is called homoderivation if h(ab) = h(a)h(b)+h(a)b+ah(b) for all a, b ∈ R. We investigate the behavior of three generalized derivations F, G, and H of R satisfying the condition

\(F\left({\xi }^{2}\right)=G\left({\xi }^{2}\right)+H\left(\xi \right)\xi +\xi H\left(\xi \right)\) $F\left({\xi }^2\right)=G\left({\xi }^2\right)+H\left(\xi \right)\xi +\xi H\left(\xi \right)$

for all ξ ∈ f(R) = {f(ξ₁, . . . , ξ_n) | ξ₁, . . . , ξ_n ∈ R}.