On generalized derivations in semiprime rings involving anticommutator

Let \(\mathfrak {R}\) be a prime ring with characteristic different from 2 and m, n, k be fixed positive integers. In this paper we study the case when \(\mathfrak {R}\) admits a generalized derivation \(\mathscr {F}\) with associated derivation \(\mathfrak {D}\) such that \((i)~(\mathscr {F}(x)\circ \mathscr {F}(y))^k=\mathscr {F}(x\circ _ky)~(ii)~ \mathscr {F}(x)\circ _m \mathscr {F}(y) =(\mathscr {F}(x\circ y))^n~(iii)~(\mathscr {F}\left( x\right) \circ \mathscr {F}\left( y\right) )^m =(\mathscr {F}(x\circ y))^n,\) for all \(x,y \in \mathscr {I}\), where \( \mathscr {I}\) is a non-zero ideal of \(\mathfrak {R}\). Moreover, we also examine the case when \(\mathfrak {R}\) is a semiprime ring.