## Higher-order linear differential equations with solutions having a prescribed sequence of zeros and lying in the Dirichlet space

### Volume 115 / 2015

Annales Polonici Mathematici 115 (2015), 275-295
MSC: Primary 34M10; Secondary 30J10.
DOI: 10.4064/ap115-3-6

#### Abstract

The aim of this paper is to consider the following three problems:

(1) for a given uniformly $q$-separated sequence satisfying certain conditions, find a coefficient function $A(z)$ analytic in the unit disc such that $f'''+A(z)f=0$ possesses a solution having zeros precisely at the points of this sequence;

(2) find necessary and sufficient conditions for the differential equation $$ f^{(k)}+A_{k-1}f^{(k-1)}+\cdots+A_1f'+A_0f=0\tag*{$(*)$} $$ in the unit disc to be Blaschke-oscillatory;(3) find sufficient conditions on the analytic coefficients of the differential equation $(*)$ for all analytic solutions to belong to the Dirichlet space $\mathcal{D}$.

Our results are a generalization of some earlier results due to J. Heittokangas and J. Gröhn.