Unbounded Fatou components of composite transcendental meromorphic functions with finitely many poles

Abstract

Let fi, i = 1, 2,..., m be transcendental meromorphic functions of order less than1 2with at most finitely many poles and at least one of them has positive lower order. Let g = fm{ring operator} fm-1{ring operator}. . .{ring operator} f1. Then either g has no unbounded Fatou components or at least one unbounded Fatou component g is multiply connected.