Complex oscillation and nonoscillation results

For an entire coefficient A(z), classifying the oscillation of solutions f of the linear differential equation f " + A(z)f = 0 has been a long-standing problem since the early 1980s. New results on the following three typical questions are proved: Under which conditions on A(z) does there exist a solution f such that (Q1) f has no zeros, (Q2)lambda(f) >= sigma(A), (Q3)lambda(f) = infinity? Here lambda(g) and sigma(g) denote the exponent of convergence of the zeros of g and the order of growth of g, respectively. Several nontrivial examples are given.