Zero distribution of solutions of some linear differential equations

Assuming that A(z) = B-1(z)e(P(z)) + B-2(z)\e(rho p(z)) + Q(z), where B-1(z), B-2(z), p(z) are nonzero polynomials, rho is an element of (0, 1), and Q(z) is an entire function of order less than deg p, we study the oscillation of solutions of the linear differential equation f((k))+A(z)f=0. Some conditions on $ A(z) $ A(z) are given to guarantee that any non-trivial solution f satisfies lambda(f) = infinity or lambda(f) >= sigma(A), where sigma(f) and lambda(f) denote respectively the order of f and the exponent of convergence of zeros of f. We also find some types of linear differential equations with zero-free solutions.