On the Value Distribution of Composite Meromorphic Functions

Let f and g be transcendental entire functions and let p be a nonconstant polynomial. A recent result of Bergweiler [2] says that the function f(g(z))− p(z) has infinitely many zeros, confirming a conjecture of Gross [11] dealing with the special case p(z) = z. The case that f(g) is of finite order follows from either of the earlier results of Gol’dberg and Prokopovich [8], Goldstein [9], Gross and Yang [15], and Mues [19]. In fact, various generalizations are obtained in these papers. In particular, it follows from each of these papers that if f and g are entire, p is a nonconstant polynomial, and f(g(z)) − p(z) has only finitely many zeros, then either f is linear or there exists a polynomial q such that p = q(g), provided f(g) is transcendental and of finite order (the restriction on the order not being essential, as shown in [4]). This latter result does not hold for meromorphic f , even if f(g) has finite order, as shown by the example f(z) = i √ z tan √ z, g(z) = z, and p(z) = z. It is natural to conjecture, however, that the function f(g(z)) − R(z) has infinitely many zeros, if f is meromorphic and transcendental, g is entire and transcendental, and R is rational and nonconstant. As proved in [3], this is in fact the case if R(z) = z and hence for any Mobius transformation R. The method used in [3], however, does not seem to be suitable to handle the case that the degree of R is greater than one. In this paper, we give an affirmative answer to the above question in the case that f(g) has finite order.