An Extension of the Corollary to Steinmetzs Theorem

In two recent papers, "A Simpler Proof of a Theorem of Steinmetz" (1989, J. Math. Anal. Appl.143, 290-294) and "An Extension of a Theorem of Steinmetz" (1991, J. Math. Anal. Appl.156, 287-292), the present authors gave a new proof of Steinmetz′s theorem that avoided use of the Second Fundamental Theorem of Nevanlinna. It was also shown in the second paper how to generalize Steinmetz′s theorem to handle the case where the inner function is not precisely the same in all cases. In this paper we shall extend these results even further.