Oscillation of fast growing solutions of linear differential equations in the unit disc

Abstract

In this paper, we investigate the relationship between solutions and their derivatives of the differential equation f (k) + A (z) f = 0, k ≥ 2, where A (z) 6≡ 0 is an analytic function with finite iterated porder and analytic functions of finite iterated p-order in the unit disc ∆ = {z ∈ C : |z| < 1}. Instead of looking at the zeros of f (j) (z) − z (j = 0, .., k), we proceed to a slight generalization by considering zeros of f (j) (z) − ϕ (z) (j = 0, .., k), where ϕ is a small analytic function relative to f such that ϕ(k−j) (z) 6≡ 0 (j = 0, ..., k), while the solution f is of infinite iterated p-order. This paper improves some very recent results of T. B. Cao and G. Zhang, A. Chen.