Abstract Versions of L′Hôpital′s 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.