ABSTRACT VERSIONS OF LHOPITALS RULE FOR HOLOMORPHIC-FUNCTIONS IN THE FRAMEWORK OF COMPLEX B-MODULES

Abstract versions of L′Hôpital′s rule are proved for the "ratio" f(z)(g(z))-1, where f: S → X, g: S → A are vector-valued holomorphic functions defined in a region of the complex plane containing S, A being a complex unilal Banach algebra, and X a complex Banach module over A. Both cases, (i) (g(z))-1[formula] 0, and (ii) f(z) [formula] 0, g(z) [formula] 0, as z[formula] α, α being either finite or infinite, are considered when f′(z)(g′(z))-1 has a finite limit. Applications are given to the asymptotics of linear second-order differential equations in Banach algebras. © 1993 Academic Press, Inc.