Let A be a ring, an additive map : is said to be a generalized derivation if there exists a derivation h of A such that satisfies (x,y. Let (A) denotes the set of all generalized derivations; if A is a normed algebra, denotes the setof all in (A) which are also bounded linear operators on A.
In this note, the author estimates the distance of the composition of two derivations , , and obtained the following results:
1. Let A be an ultraprime normed algebra and let then if (a,b, where ,
2. Let A be an ultrasemiprime normed algebra and , then if satisfies for all
3. Let A be a von Neumann algebra. If then . In particular, for any ,
As a consequence of these results, the author obtains a partial answer to Mathieu’s question; if then .