On the zeros of solutions of linear differential equations of the second order

Let u be a solution of the differential equation u " + Ru = 0, where R is r ational. Newton's method of finding the zeros of u consists of iterating th e function f(z) = z - u(z)/u'(z). With suitable hypotheses on R and a, it i s shown that the iterates off converge on an open dense subset of the plane if they converge for the zeros of R. The proof is based on the iteration t heory of meromorphic functions,and in particular on the result that, if the family of K-quasiconformal deformations of a meromorphic function f depend s on only finitely many parameters, then every cycle of Baker domains off c ontains a singularity of f(-1). This result, together with classical result s of Hille concerning the asymptotic behaviour of solutions of the above di fferential equations, is also used to study their value distribution. For e xample, it is shown that, if R is a rational function which satisfies R(z) similar to a(m)z(m) as z --> infinity and has only k distinct zeros where k < (m + 2)/2, then delta(0, u) less than or equal to k/(m + 2 - k) < 1.