Zero distribution of solutions of some linear differential equations

Assuming that A(z)=B1(z)eP(z)+B2(z)eϱp(z)+Q(z), where B1(z),B2(z),p(z) are nonzero polynomials, ϱ∈(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) are given to guarantee that any non-trivial solution f satisfies λ(f)=∞ or λ(f)≥σ(A), where σ(f) and λ(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.