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

Let f(i), i = 1, 2, ..., m be transcendental meromorphic functions of order less than 1/2 with at most finitely many poles and at least one of them has positive lower order. Let g = f(m) circle f(m-1) circle ... circle f(1). Then either g has no unbounded Fatou components or at least one unbounded Fatou component g is multiply connected.