Invertibility of Holomorphic Functions with Respect to the Hadamard Product *

Abstract

Let G 1 and G 2 be domains in containing the open unit disk D, and let H(Gj ) be the set of all holomorphic functions in Gj for j = 1,2. We say that f εH(G 1) is invertible with respect to H(G 2) if there exists g ε H(G 2) such that (f * g)(z)= - 1/(1 - z) for z ε D, where f*g denotes the Hadamard product of f and g. In (R. Brück and J. Müller (1992). Math. Ann., 294, 421–438) it was shown that for certain kinds of domains G 1 and G 2 the invertible elements of H(G 1) with respect to H(G 2) are necessarily of a very special form. The main purpose of this paper is to show that this result is true for broader classes of domains, where our method of proof is different to that in (R. Brück and J. Müller (1992). Math. Ann., 294, 421-438).