On the Meromorphic Solutions of Non-Homogeneous Linear Differential Equations with Polynomial Coefficients

A lower bound for the number of zeros of a meromorphic function and its second derivative

“…We state the following result, part (i) of which was proved by Frank and Hellerstein in [6], with part (ii) appearing in outside an exceptional set. With the hypotheses of Theorem D, part (ii), the assumption that H is transcendental and (1.2) lead to an upper estimate for m(r, 1//), which in turn contradicts the fact that N(r, 1//) is small.…”

“…The proof of Theorem A in [12] begins by using a device of Frank [5,6,7]. If f u f 2 are linearly independent solutions of the associated homogeneous equation w" + a x w' + a o w = 0, then we define g by g 2 =f/F, and the functions Wj=f'jg-fj(f'/f)g solve an equation w" + a 1 w' + bw = 0 in which b is meromorphic with only finitely many poles.…”

On the zeros of second order linear differential polynomials

“…It is natural to extend the above problem to consideration of the zeros of a meromorphic function /(z) and a linear differential polynomial F in f(z), that is < k > I aj(z)f^(z) (1.1) where the a, are, say, rational. Among other results in [2], Frank and Hellerstein classified completely those entire functions f(z) such that f(z) and F{z) have only finitely many zeros, where F is given by (1.1) with fc^2 and the a,-polynomials. They also showed that if / is meromorphic in the plane, and / and F have only finitely many zeros, where /c^3 and the a,-are again polynomials, then / ' / / has finite order determined by the degrees of the a y For constant coefficients, Steinmetz Here a, b, c, d are constants and n is a positive integer.…”

“…Assume for the time being that Re(l-2v)#0 and that G^G^O in S 2 , that is that Re (1 -2v) > 0. At a large pole of / in S 2 , H = 1 gives using (4.27).…”

“…Suppose further that N(r,f) has finite lower order. By results from [2], where we apply Theorem 1 if / is entire and Lemma 8 if / is not entire, / ' / / has finite lower order. We shall prove that either / is given by (1.2), or / has finite order, so that in the latter case we can appeal to Theorem C. Clearly we may assume that <x = 1, that / is transcendental, and that F is non-constant.…”