Zero Distribution of Solutions of Higher-Order Linear Differential Equations and Zygmund Type Space

Abstract:

      The aim of this paper is to consider the following two problems: (1)~~Establish interrelationships between the growth of coefficients and the geometric distribution of zeros of solutions of non-homogeneous linear differential equation $$f'''+A_2(z)f''+A_1(z)f'+A_0(z)f=A_3(z),$$ where $A_0(z),\ldots, A_3(z)$ are analytic functions in the unit disc $\mathbb{D}$; (2)~~Find some sufficient conditions on the analytic coefficients of the differential equation $$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_1(z)f'+A_0(z)f=0,$$ for all solutions to belong to the Zygmund type space. The results we obtain are a generalization of some earlier results by Heittokangas, Gr\"{o}hn, Korhoneon and R\"{a}tty\"{a}.