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Let [Formula: see text] be a unital prime Banach algebra over complex field [Formula: see text] with unity and [Formula: see text] be a nonzero continuous linear generalized derivation associated with a nonzero continuous linear derivation [Formula: see text]. In this paper, we investigate the commutativity of [Formula: see text]. In particular, we prove that a unital prime Banach algebra [Formula: see text] is commutative if one of the following holds; (i) either [Formula: see text] or [Formula: see text], (ii) either [Formula: see text] or [Formula: see text], for sufficiently many [Formula: see text], for any complex numbers [Formula: see text] and an integer [Formula: see text].

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Let R be a prime ring of characteristic different from 2 and F a b-generalized derivation on R. Let U be Utumi quotient ring of R with extended centroid C and f (x 1 ,. .. , x n) be a multilinear polynomial over C which is not central valued on R. Suppose that d is a non zero derivation on R such that d([F(f (r)), f (r)]) ∈ C for all r = (r 1 ,. .. , r n) ∈ R n ; then one of the following holds: (1) there exist a ∈ U, λ ∈ C such that F(x) = ax + λx + xa for all x ∈ R and f (x 1 ,. .. , x n) 2 is central valued on R, (2) there exists λ ∈ C such that F(x) = λx for all x ∈ R.