On the Zeros of Linear Differential Polynomials with Small Rational Coefficients

We prove the following: suppose that J{z) is transcendental and meromorphic of finite order in the plane, and that the linear differential polynomial F(z) is defined by and is non-constant, where ak_j(z), ...,ao(z) are rational functions vanishing at infinity. Then N(r,l/(fFF')) = implies that N(r,f) = O(logr). A corresponding result is proved for the case where F =f' + af, where a is a constant. The problem is related to results of Frank and Hellerstein and others.