A new proof of Singer-Wermer Theorem with some results on {g, h}-derivations.

Singer and Wermer proved that if A is a commutative Banach algebra and d: A → A is a continuous derivation, then d(A) ⊆ rad(A), where rad(A) denotes the Jacobson radical of A. In this paper, we establish a new proof of that theorem. Moreover, we prove that every continuous Jordan derivation on a finite dimensional Banach algebra, under certain conditions, is identically zero. As another objective of this article, we study {g, h}-derivations on algebras. In this regard, we prove that if f is a {g, h}-derivation on a unital algebra, then f, g and h are generalized derivations. Additionally, we achieve some results concerning the automatic continuity of {g, h}-derivations on Banach algebras. In the last section of the article, we introduce the concept of a {g, h}-homomorphism and then we present a characterization of it under certain conditions.