Power values of cocentralizing b-generalized derivations of prime rings

Let [Formula: see text] be a prime ring, [Formula: see text] the right maximal ring of quotients of [Formula: see text], [Formula: see text] the extended centroid of [Formula: see text], [Formula: see text] a non-central Lie ideal of [Formula: see text], [Formula: see text] and [Formula: see text] are non-zero [Formula: see text]-generalized derivations of [Formula: see text]. Suppose that there exists a fixed positive integer [Formula: see text] such that [Formula: see text] for all [Formula: see text]. Then one of the following holds: (i) [Formula: see text] satisfies [Formula: see text] and [Formula: see text]; (ii) There exists some [Formula: see text] such that [Formula: see text] for all [Formula: see text]; (iii) There exists some [Formula: see text] such that [Formula: see text] for all [Formula: see text]; (iv) [Formula: see text] satisfies [Formula: see text] and there exist [Formula: see text] such that [Formula: see text] for all [Formula: see text] with [Formula: see text].